Theorem reu2eqd 3708

Description: Deduce equality from restricted uniqueness, deduction version. (Contributed by Thierry Arnoux, 27-Nov-2019.)

Hypotheses

Ref	Expression
reu2eqd.1	⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
reu2eqd.2	⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜃))
reu2eqd.3	⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
reu2eqd.4	⊢ (𝜑 → 𝐵 ∈ 𝐴)
reu2eqd.5	⊢ (𝜑 → 𝐶 ∈ 𝐴)
reu2eqd.6	⊢ (𝜑 → 𝜒)
reu2eqd.7	⊢ (𝜑 → 𝜃)

Assertion

Ref	Expression
reu2eqd	⊢ (𝜑 → 𝐵 = 𝐶)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝜒,𝑥   𝜃,𝑥

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem reu2eqd

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reu2eqd.6	. 2 ⊢ (𝜑 → 𝜒)
2		reu2eqd.7	. 2 ⊢ (𝜑 → 𝜃)
3		reu2eqd.3	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 𝜓)
4		reu2 3697	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ (∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
5	3, 4	sylib 221	. . . 4 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦)))
6	5	simprd 500	. . 3 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦))
7		reu2eqd.4	. . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
8		reu2eqd.5	. . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝐴)
9		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑥𝜒
10		nfs1v 2197	. . . . . . 7 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓
11	9, 10	nfan 1926	. . . . . 6 ⊢ Ⅎ𝑥(𝜒 ∧ [𝑦 / 𝑥]𝜓)
12		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑥 𝐵 = 𝑦
13	11, 12	nfim 1923	. . . . 5 ⊢ Ⅎ𝑥((𝜒 ∧ [𝑦 / 𝑥]𝜓) → 𝐵 = 𝑦)
14		nfv 1941	. . . . 5 ⊢ Ⅎ𝑦((𝜒 ∧ 𝜃) → 𝐵 = 𝐶)
15		reu2eqd.1	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝜓 ↔ 𝜒))
16	15	anbi1d 642	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝜓 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜒 ∧ [𝑦 / 𝑥]𝜓)))
17		eqeq1 2773	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥 = 𝑦 ↔ 𝐵 = 𝑦))
18	16, 17	imbi12d 347	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) ↔ ((𝜒 ∧ [𝑦 / 𝑥]𝜓) → 𝐵 = 𝑦)))
19		nfv 1941	. . . . . . . 8 ⊢ Ⅎ𝑥𝜃
20		reu2eqd.2	. . . . . . . 8 ⊢ (𝑥 = 𝐶 → (𝜓 ↔ 𝜃))
21	19, 20	sbhypf 3522	. . . . . . 7 ⊢ (𝑦 = 𝐶 → ([𝑦 / 𝑥]𝜓 ↔ 𝜃))
22	21	anbi2d 641	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝜒 ∧ [𝑦 / 𝑥]𝜓) ↔ (𝜒 ∧ 𝜃)))
23		eqeq2 2781	. . . . . 6 ⊢ (𝑦 = 𝐶 → (𝐵 = 𝑦 ↔ 𝐵 = 𝐶))
24	22, 23	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝜒 ∧ [𝑦 / 𝑥]𝜓) → 𝐵 = 𝑦) ↔ ((𝜒 ∧ 𝜃) → 𝐵 = 𝐶)))
25	13, 14, 18, 24	rspc2 3599	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜒 ∧ 𝜃) → 𝐵 = 𝐶)))
26	7, 8, 25	syl2anc 595	. . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜓 ∧ [𝑦 / 𝑥]𝜓) → 𝑥 = 𝑦) → ((𝜒 ∧ 𝜃) → 𝐵 = 𝐶)))
27	6, 26	mpd 16	. 2 ⊢ (𝜑 → ((𝜒 ∧ 𝜃) → 𝐵 = 𝐶))
28	1, 2, 27	mp2and 711	1 ⊢ (𝜑 → 𝐵 = 𝐶)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  [wsb 2097   ∈ wcel 2149  ∀wral 3085  ∃wrex 3095  ∃!wreu 3374

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-reu 3377

This theorem is referenced by:  qtophmeo  23942  footeq  28962  mideulem2  28973  lmieq  29057  fineqvnttrclse  35459  upciclem3  49830