Theorem csbied2 2949

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
csbied2.1	|- ( ph -> A e. V )
csbied2.2	|- ( ph -> A = B )
csbied2.3	|- ( ( ph /\ x = B ) -> C = D )

Assertion

Ref	Expression
csbied2	|- ( ph -> [_ A / x ]_ C = D )

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

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

Proof of Theorem csbied2

Step	Hyp	Ref	Expression
1		csbied2.1	. 2 |- ( ph -> A e. V )
2		id 19	. . . 4 |- ( x = A -> x = A )
3		csbied2.2	. . . 4 |- ( ph -> A = B )
4	2, 3	sylan9eqr 2135	. . 3 |- ( ( ph /\ x = A ) -> x = B )
5		csbied2.3	. . 3 |- ( ( ph /\ x = B ) -> C = D )
6	4, 5	syldan 276	. 2 |- ( ( ph /\ x = A ) -> C = D )
7	1, 6	csbied 2948	1 |- ( ph -> [_ A / x ]_ C = D )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   [_csb 2908

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-sbc 2816  df-csb 2909

This theorem is referenced by: (None)