Theorem cbvcsbv 3141

Description: Change the bound variable of a proper substitution into a class using implicit substitution. (Contributed by NM, 30-Sep-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypothesis

Ref	Expression
cbvcsbv.1	⊢ (x = y → B = C)

Assertion

Ref	Expression
cbvcsbv	⊢ [A / x]B = [A / y]C

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

Allowed substitution hints:   A(x,y)   B(x)   C(y)

Proof of Theorem cbvcsbv

Step	Hyp	Ref	Expression
1		nfcv 2489	. 2 ⊢ ℲyB
2		nfcv 2489	. 2 ⊢ ℲxC
3		cbvcsbv.1	. 2 ⊢ (x = y → B = C)
4	1, 2, 3	cbvcsb 3140	1 ⊢ [A / x]B = [A / y]C

Syntax hints:   → wi 4   = wceq 1642  [csb 3136

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 2478  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)