Theorem riotasvd 36084

Description: Deduction version of riotasv 36087. (Contributed by NM, 4-Mar-2013.) (Revised by Mario Carneiro, 15-Oct-2016.)

Hypotheses

Ref	Expression
riotasvd.1	⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
riotasvd.2	⊢ (𝜑 → 𝐷 ∈ 𝐴)

Assertion

Ref	Expression
riotasvd	⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵   𝑥,𝐶   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑦)   𝐵(𝑦)   𝐶(𝑦)   𝐷(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem riotasvd

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		riotasvd.1	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
2	1	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)))
3		riotasvd.2	. . . . . . . . 9 ⊢ (𝜑 → 𝐷 ∈ 𝐴)
4	3	adantr 483	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → 𝐷 ∈ 𝐴)
5	2, 4	eqeltrrd 2912	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴)
6		riotaclbgBAD 36082	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
7	6	adantl 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴))
8	5, 7	mpbird 259	. . . . . 6 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
9		riotasbc 7124	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) → [(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
10	8, 9	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → [(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
11		eqeq1 2823	. . . . . . . . 9 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐶 ↔ 𝑧 = 𝐶))
12	11	imbi2d 343	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → ((𝜓 → 𝑥 = 𝐶) ↔ (𝜓 → 𝑧 = 𝐶)))
13	12	ralbidv 3195	. . . . . . 7 ⊢ (𝑥 = 𝑧 → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶)))
14		nfra1 3217	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)
15		nfcv 2975	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝐴
16	14, 15	nfriota 7118	. . . . . . . . 9 ⊢ Ⅎ𝑦(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
17	16	nfeq2 2993	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶))
18		eqeq1 2823	. . . . . . . . 9 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (𝑧 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
19	18	imbi2d 343	. . . . . . . 8 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → ((𝜓 → 𝑧 = 𝐶) ↔ (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
20	17, 19	ralbid 3229	. . . . . . 7 ⊢ (𝑧 = (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) → (∀𝑦 ∈ 𝐵 (𝜓 → 𝑧 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
21	13, 20	sbcie2g 3809	. . . . . 6 ⊢ ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) ∈ 𝐴 → ([(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
22	5, 21	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ([(℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) / 𝑥]∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶) ↔ ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
23	10, 22	mpbid 234	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
24		rsp 3203	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶) → (𝑦 ∈ 𝐵 → (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
25	23, 24	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝑦 ∈ 𝐵 → (𝜓 → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶)))
26	25	impd 413	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
27	2	eqeq1d 2821	. 2 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝐷 = 𝐶 ↔ (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 (𝜓 → 𝑥 = 𝐶)) = 𝐶))
28	26, 27	sylibrd 261	1 ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → ((𝑦 ∈ 𝐵 ∧ 𝜓) → 𝐷 = 𝐶))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1531   ∈ wcel 2108  ∀wral 3136  ∃!wreu 3138  [wsbc 3770  ℩crio 7105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-riota 7106  df-undef 7931

This theorem is referenced by:  riotasv2d  36085  riotasv  36087  riotasv3d  36088  cdleme32a  37569