Theorem cbvcsb 3141

Description: Change bound variables in a class substitution. Interestingly, this does not require any bound variable conditions on A. (Contributed by Jeff Hankins, 13-Sep-2009.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
cbvcsb.1	⊢ ℲyC
cbvcsb.2	⊢ ℲxD
cbvcsb.3	⊢ (x = y → C = D)

Assertion

Ref	Expression
cbvcsb	⊢ [A / x]C = [A / y]D

Proof of Theorem cbvcsb

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvcsb.1	. . . . 5 ⊢ ℲyC
2	1	nfcri 2484	. . . 4 ⊢ Ⅎy z ∈ C
3		cbvcsb.2	. . . . 5 ⊢ ℲxD
4	3	nfcri 2484	. . . 4 ⊢ Ⅎx z ∈ D
5		cbvcsb.3	. . . . 5 ⊢ (x = y → C = D)
6	5	eleq2d 2420	. . . 4 ⊢ (x = y → (z ∈ C ↔ z ∈ D))
7	2, 4, 6	cbvsbc 3075	. . 3 ⊢ ([̣A / x]̣z ∈ C ↔ [̣A / y]̣z ∈ D)
8	7	abbii 2466	. 2 ⊢ {z ∣ [̣A / x]̣z ∈ C} = {z ∣ [̣A / y]̣z ∈ D}
9		df-csb 3138	. 2 ⊢ [A / x]C = {z ∣ [̣A / x]̣z ∈ C}
10		df-csb 3138	. 2 ⊢ [A / y]D = {z ∣ [̣A / y]̣z ∈ D}
11	8, 9, 10	3eqtr4i 2383	1 ⊢ [A / x]C = [A / y]D

Syntax hints:   → wi 4   = wceq 1642   ∈ wcel 1710  {cab 2339  Ⅎwnfc 2477  [̣wsbc 3047  [csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-sbc 3048  df-csb 3138

This theorem is referenced by:  cbvcsbv  3142