Theorem cbvcsbv 3897

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 2979	. 2 ⊢ Ⅎ𝑦𝐵
2		nfcv 2979	. 2 ⊢ Ⅎ𝑥𝐶
3		cbvcsbv.1	. 2 ⊢ (𝑥 = 𝑦 → 𝐵 = 𝐶)
4	1, 2, 3	cbvcsbw 3895	1 ⊢ ⦋𝐴 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑦⦌𝐶

Syntax hints:   → wi 4   = wceq 1537  ⦋csb 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-sbc 3775  df-csb 3886

This theorem is referenced by:  pmatcollpw3lem  21393  poimirlem27  34921  cdleme40v  37607