Theorem csbiotag 4895

Description: Class substitution within a description binder. (Contributed by Scott Fenton, 6-Oct-2017.)

Assertion

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

Distinct variable groups:   𝑦,𝐴   𝑥,𝑦

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

Proof of Theorem csbiotag

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 2855	. . 3 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑))
2		dfsbcq2 2767	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	iotabidv 4888	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2054	. 2 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5		vex 2560	. . 3 ⊢ 𝑧 ∈ V
6		nfs1v 1815	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	6	nfiotaxy 4871	. . 3 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8		sbequ12 1654	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	8	iotabidv 4888	. . 3 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
10	5, 7, 9	csbief 2891	. 2 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
11	4, 10	vtoclg 2613	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  [wsb 1645  [wsbc 2764  ⦋csb 2852  ℩cio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-sn 3381  df-uni 3581  df-iota 4867

This theorem is referenced by:  csbfv12g  5209