Theorem csbriotag 5846

Description: Interchange class substitution and restricted description binder. (Contributed by NM, 24-Feb-2013.)

Assertion

Ref	Expression
csbriotag	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥)   𝐵(𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem csbriotag

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3062	. . 3 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑))
2		dfsbcq2 2967	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	riotabidv 5836	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2192	. 2 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑)))
5		vex 2742	. . 3 ⊢ 𝑧 ∈ V
6		nfs1v 1939	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7		nfcv 2319	. . . 4 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfriota 5843	. . 3 ⊢ Ⅎ𝑥(℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
9		sbequ12 1771	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
10	9	riotabidv 5836	. . 3 ⊢ (𝑥 = 𝑧 → (℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑))
11	5, 8, 10	csbief 3103	. 2 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝑧 / 𝑥]𝜑)
12	4, 11	vtoclg 2799	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦 ∈ 𝐵 𝜑) = (℩𝑦 ∈ 𝐵 [𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1353  [wsb 1762   ∈ wcel 2148  [wsbc 2964  ⦋csb 3059  ℩crio 5833

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2741  df-sbc 2965  df-csb 3060  df-sn 3600  df-uni 3812  df-iota 5180  df-riota 5834

This theorem is referenced by: (None)