Theorem csbiotag 5124

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 3010	. . 3 ⊢ (𝑧 = 𝐴 → ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = ⦋𝐴 / 𝑥⦌(℩𝑦𝜑))
2		dfsbcq2 2916	. . . 4 ⊢ (𝑧 = 𝐴 → ([𝑧 / 𝑥]𝜑 ↔ [𝐴 / 𝑥]𝜑))
3	2	iotabidv 5117	. . 3 ⊢ (𝑧 = 𝐴 → (℩𝑦[𝑧 / 𝑥]𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))
4	1, 3	eqeq12d 2155	. 2 ⊢ (𝑧 = 𝐴 → (⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑) ↔ ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑)))
5		vex 2692	. . 3 ⊢ 𝑧 ∈ V
6		nfs1v 1913	. . . 4 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
7	6	nfiotaw 5100	. . 3 ⊢ Ⅎ𝑥(℩𝑦[𝑧 / 𝑥]𝜑)
8		sbequ12 1745	. . . 4 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
9	8	iotabidv 5117	. . 3 ⊢ (𝑥 = 𝑧 → (℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑))
10	5, 7, 9	csbief 3049	. 2 ⊢ ⦋𝑧 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝑧 / 𝑥]𝜑)
11	4, 10	vtoclg 2749	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(℩𝑦𝜑) = (℩𝑦[𝐴 / 𝑥]𝜑))

Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 1481  [wsb 1736  [wsbc 2913  ⦋csb 3007  ℩cio 5094

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-rex 2423  df-v 2691  df-sbc 2914  df-csb 3008  df-sn 3538  df-uni 3745  df-iota 5096

This theorem is referenced by:  csbfv12g  5465