Theorem ifeqeqx 30065

Description: An equality theorem tailored for ballotlemsf1o 31423. (Contributed by Thierry Arnoux, 14-Apr-2017.)

Hypotheses

Ref	Expression
ifeqeqx.1	⊢ (𝑥 = 𝑋 → 𝐴 = 𝐶)
ifeqeqx.2	⊢ (𝑥 = 𝑌 → 𝐵 = 𝑎)
ifeqeqx.3	⊢ (𝑥 = 𝑋 → (𝜒 ↔ 𝜃))
ifeqeqx.4	⊢ (𝑥 = 𝑌 → (𝜒 ↔ 𝜓))
ifeqeqx.5	⊢ (𝜑 → 𝑎 = 𝐶)
ifeqeqx.6	⊢ ((𝜑 ∧ 𝜓) → 𝜃)
ifeqeqx.y	⊢ (𝜑 → 𝑌 ∈ 𝑉)
ifeqeqx.x	⊢ (𝜑 → 𝑋 ∈ 𝑊)

Assertion

Ref	Expression
ifeqeqx	⊢ ((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))

Distinct variable groups:   𝑥,𝑎   𝑥,𝐶   𝑥,𝑋   𝑥,𝑌   𝑥,𝑉   𝑥,𝑊   𝜓,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑎)   𝜓(𝑎)   𝜒(𝑥,𝑎)   𝜃(𝑎)   𝐴(𝑥,𝑎)   𝐵(𝑥,𝑎)   𝐶(𝑎)   𝑉(𝑎)   𝑊(𝑎)   𝑋(𝑎)   𝑌(𝑎)

Proof of Theorem ifeqeqx

Step	Hyp	Ref	Expression
1		eqeq2 2789	. 2 ⊢ (𝐴 = if(𝜒, 𝐴, 𝐵) → (𝑎 = 𝐴 ↔ 𝑎 = if(𝜒, 𝐴, 𝐵)))
2		eqeq2 2789	. 2 ⊢ (𝐵 = if(𝜒, 𝐴, 𝐵) → (𝑎 = 𝐵 ↔ 𝑎 = if(𝜒, 𝐴, 𝐵)))
3		simplr 756	. . 3 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ 𝜒) → 𝑥 = if(𝜓, 𝑋, 𝑌))
4		simpll 754	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ 𝜒) → 𝜑)
5		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ 𝜒) → 𝜒)
6		sbceq1a 3692	. . . . . 6 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (𝜒 ↔ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
7	6	biimpd 221	. . . . 5 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (𝜒 → [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
8	3, 5, 7	sylc 65	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ 𝜒) → [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒)
9		dfsbcq 3683	. . . . . 6 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → ([𝑋 / 𝑥]𝜒 ↔ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
10		csbeq1 3789	. . . . . . 7 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → ⦋𝑋 / 𝑥⦌𝐴 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴)
11	10	eqeq2d 2788	. . . . . 6 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → (𝑎 = ⦋𝑋 / 𝑥⦌𝐴 ↔ 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴))
12	9, 11	imbi12d 337	. . . . 5 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → (([𝑋 / 𝑥]𝜒 → 𝑎 = ⦋𝑋 / 𝑥⦌𝐴) ↔ ([if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒 → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴)))
13		dfsbcq 3683	. . . . . 6 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → ([𝑌 / 𝑥]𝜒 ↔ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
14		csbeq1 3789	. . . . . . 7 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → ⦋𝑌 / 𝑥⦌𝐴 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴)
15	14	eqeq2d 2788	. . . . . 6 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → (𝑎 = ⦋𝑌 / 𝑥⦌𝐴 ↔ 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴))
16	13, 15	imbi12d 337	. . . . 5 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → (([𝑌 / 𝑥]𝜒 → 𝑎 = ⦋𝑌 / 𝑥⦌𝐴) ↔ ([if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒 → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴)))
17		ifeqeqx.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝑊)
18		nfcvd 2933	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝑊 → Ⅎ𝑥𝐶)
19		ifeqeqx.1	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → 𝐴 = 𝐶)
20	18, 19	csbiegf 3812	. . . . . . . . . 10 ⊢ (𝑋 ∈ 𝑊 → ⦋𝑋 / 𝑥⦌𝐴 = 𝐶)
21	17, 20	syl 17	. . . . . . . . 9 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌𝐴 = 𝐶)
22		ifeqeqx.5	. . . . . . . . 9 ⊢ (𝜑 → 𝑎 = 𝐶)
23	21, 22	eqtr4d 2817	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑋 / 𝑥⦌𝐴 = 𝑎)
24	23	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ 𝜓) → ⦋𝑋 / 𝑥⦌𝐴 = 𝑎)
25	24	eqcomd 2784	. . . . . 6 ⊢ ((𝜑 ∧ 𝜓) → 𝑎 = ⦋𝑋 / 𝑥⦌𝐴)
26	25	a1d 25	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → ([𝑋 / 𝑥]𝜒 → 𝑎 = ⦋𝑋 / 𝑥⦌𝐴))
27		pm3.24 394	. . . . . . . . . 10 ⊢ ¬ (𝜓 ∧ ¬ 𝜓)
28		ifeqeqx.y	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑌 ∈ 𝑉)
29		ifeqeqx.4	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑌 → (𝜒 ↔ 𝜓))
30	29	sbcieg 3714	. . . . . . . . . . . 12 ⊢ (𝑌 ∈ 𝑉 → ([𝑌 / 𝑥]𝜒 ↔ 𝜓))
31	28, 30	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → ([𝑌 / 𝑥]𝜒 ↔ 𝜓))
32	31	anbi1d 620	. . . . . . . . . 10 ⊢ (𝜑 → (([𝑌 / 𝑥]𝜒 ∧ ¬ 𝜓) ↔ (𝜓 ∧ ¬ 𝜓)))
33	27, 32	mtbiri 319	. . . . . . . . 9 ⊢ (𝜑 → ¬ ([𝑌 / 𝑥]𝜒 ∧ ¬ 𝜓))
34	33	pm2.21d 119	. . . . . . . 8 ⊢ (𝜑 → (([𝑌 / 𝑥]𝜒 ∧ ¬ 𝜓) → 𝑎 = ⦋𝑌 / 𝑥⦌𝐴))
35	34	imp 398	. . . . . . 7 ⊢ ((𝜑 ∧ ([𝑌 / 𝑥]𝜒 ∧ ¬ 𝜓)) → 𝑎 = ⦋𝑌 / 𝑥⦌𝐴)
36	35	anass1rs 642	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝜓) ∧ [𝑌 / 𝑥]𝜒) → 𝑎 = ⦋𝑌 / 𝑥⦌𝐴)
37	36	ex 405	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜓) → ([𝑌 / 𝑥]𝜒 → 𝑎 = ⦋𝑌 / 𝑥⦌𝐴))
38	12, 16, 26, 37	ifbothda 4387	. . . 4 ⊢ (𝜑 → ([if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒 → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴))
39	4, 8, 38	sylc 65	. . 3 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ 𝜒) → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴)
40		csbeq1a 3795	. . . . 5 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → 𝐴 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴)
41	40	eqeq2d 2788	. . . 4 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (𝑎 = 𝐴 ↔ 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴))
42	41	biimprd 240	. . 3 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐴 → 𝑎 = 𝐴))
43	3, 39, 42	sylc 65	. 2 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ 𝜒) → 𝑎 = 𝐴)
44		simplr 756	. . 3 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ ¬ 𝜒) → 𝑥 = if(𝜓, 𝑋, 𝑌))
45		simpll 754	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ ¬ 𝜒) → 𝜑)
46		simpr 477	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ ¬ 𝜒) → ¬ 𝜒)
47	6	notbid 310	. . . . . 6 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (¬ 𝜒 ↔ ¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
48	47	biimpd 221	. . . . 5 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (¬ 𝜒 → ¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
49	44, 46, 48	sylc 65	. . . 4 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ ¬ 𝜒) → ¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒)
50	9	notbid 310	. . . . . 6 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → (¬ [𝑋 / 𝑥]𝜒 ↔ ¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
51		csbeq1 3789	. . . . . . 7 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → ⦋𝑋 / 𝑥⦌𝐵 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵)
52	51	eqeq2d 2788	. . . . . 6 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → (𝑎 = ⦋𝑋 / 𝑥⦌𝐵 ↔ 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵))
53	50, 52	imbi12d 337	. . . . 5 ⊢ (𝑋 = if(𝜓, 𝑋, 𝑌) → ((¬ [𝑋 / 𝑥]𝜒 → 𝑎 = ⦋𝑋 / 𝑥⦌𝐵) ↔ (¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒 → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵)))
54	13	notbid 310	. . . . . 6 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → (¬ [𝑌 / 𝑥]𝜒 ↔ ¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒))
55		csbeq1 3789	. . . . . . 7 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → ⦋𝑌 / 𝑥⦌𝐵 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵)
56	55	eqeq2d 2788	. . . . . 6 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → (𝑎 = ⦋𝑌 / 𝑥⦌𝐵 ↔ 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵))
57	54, 56	imbi12d 337	. . . . 5 ⊢ (𝑌 = if(𝜓, 𝑋, 𝑌) → ((¬ [𝑌 / 𝑥]𝜒 → 𝑎 = ⦋𝑌 / 𝑥⦌𝐵) ↔ (¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒 → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵)))
58		ifeqeqx.3	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑋 → (𝜒 ↔ 𝜃))
59	58	sbcieg 3714	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝑊 → ([𝑋 / 𝑥]𝜒 ↔ 𝜃))
60	17, 59	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → ([𝑋 / 𝑥]𝜒 ↔ 𝜃))
61	60	notbid 310	. . . . . . . . . . 11 ⊢ (𝜑 → (¬ [𝑋 / 𝑥]𝜒 ↔ ¬ 𝜃))
62	61	biimpd 221	. . . . . . . . . 10 ⊢ (𝜑 → (¬ [𝑋 / 𝑥]𝜒 → ¬ 𝜃))
63		ifeqeqx.6	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝜓) → 𝜃)
64	63	ex 405	. . . . . . . . . 10 ⊢ (𝜑 → (𝜓 → 𝜃))
65	62, 64	nsyld 156	. . . . . . . . 9 ⊢ (𝜑 → (¬ [𝑋 / 𝑥]𝜒 → ¬ 𝜓))
66	65	anim2d 602	. . . . . . . 8 ⊢ (𝜑 → ((𝜓 ∧ ¬ [𝑋 / 𝑥]𝜒) → (𝜓 ∧ ¬ 𝜓)))
67	27, 66	mtoi 191	. . . . . . 7 ⊢ (𝜑 → ¬ (𝜓 ∧ ¬ [𝑋 / 𝑥]𝜒))
68	67	pm2.21d 119	. . . . . 6 ⊢ (𝜑 → ((𝜓 ∧ ¬ [𝑋 / 𝑥]𝜒) → 𝑎 = ⦋𝑋 / 𝑥⦌𝐵))
69	68	expdimp 445	. . . . 5 ⊢ ((𝜑 ∧ 𝜓) → (¬ [𝑋 / 𝑥]𝜒 → 𝑎 = ⦋𝑋 / 𝑥⦌𝐵))
70		nfcvd 2933	. . . . . . . . . 10 ⊢ (𝑌 ∈ 𝑉 → Ⅎ𝑥𝑎)
71		ifeqeqx.2	. . . . . . . . . 10 ⊢ (𝑥 = 𝑌 → 𝐵 = 𝑎)
72	70, 71	csbiegf 3812	. . . . . . . . 9 ⊢ (𝑌 ∈ 𝑉 → ⦋𝑌 / 𝑥⦌𝐵 = 𝑎)
73	28, 72	syl 17	. . . . . . . 8 ⊢ (𝜑 → ⦋𝑌 / 𝑥⦌𝐵 = 𝑎)
74	73	adantr 473	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝜓) → ⦋𝑌 / 𝑥⦌𝐵 = 𝑎)
75	74	eqcomd 2784	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝜓) → 𝑎 = ⦋𝑌 / 𝑥⦌𝐵)
76	75	a1d 25	. . . . 5 ⊢ ((𝜑 ∧ ¬ 𝜓) → (¬ [𝑌 / 𝑥]𝜒 → 𝑎 = ⦋𝑌 / 𝑥⦌𝐵))
77	53, 57, 69, 76	ifbothda 4387	. . . 4 ⊢ (𝜑 → (¬ [if(𝜓, 𝑋, 𝑌) / 𝑥]𝜒 → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵))
78	45, 49, 77	sylc 65	. . 3 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ ¬ 𝜒) → 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵)
79		csbeq1a 3795	. . . . 5 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → 𝐵 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵)
80	79	eqeq2d 2788	. . . 4 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (𝑎 = 𝐵 ↔ 𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵))
81	80	biimprd 240	. . 3 ⊢ (𝑥 = if(𝜓, 𝑋, 𝑌) → (𝑎 = ⦋if(𝜓, 𝑋, 𝑌) / 𝑥⦌𝐵 → 𝑎 = 𝐵))
82	44, 78, 81	sylc 65	. 2 ⊢ (((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) ∧ ¬ 𝜒) → 𝑎 = 𝐵)
83	1, 2, 43, 82	ifbothda 4387	1 ⊢ ((𝜑 ∧ 𝑥 = if(𝜓, 𝑋, 𝑌)) → 𝑎 = if(𝜒, 𝐴, 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  [wsbc 3681  ⦋csb 3786  ifcif 4350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-v 3417  df-sbc 3682  df-csb 3787  df-if 4351

This theorem is referenced by:  ballotlemsf1o  31423