Theorem csbied 2892

Description: Conversion of implicit substitution to explicit substitution into a class. (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
csbied.1	|- ( ph -> A e. V )
csbied.2	|- ( ( ph /\ x = A ) -> B = C )

Assertion

Ref	Expression
csbied	|- ( ph -> [_ A / x ]_ B = C )

Distinct variable groups:   x, A   x, C   ph, x

Allowed substitution hints:   B( x)   V( x)

Proof of Theorem csbied

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ x ph
2		nfcvd 2179	. 2 |- ( ph -> F/_ x C )
3		csbied.1	. 2 |- ( ph -> A e. V )
4		csbied.2	. 2 |- ( ( ph /\ x = A ) -> B = C )
5	1, 2, 3, 4	csbiedf 2887	1 |- ( ph -> [_ A / x ]_ B = C )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   [_csb 2852

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-v 2559  df-sbc 2765  df-csb 2853

This theorem is referenced by:  csbied2  2893  fvmptd  5253