Theorem csbied2 3092

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 2221	. . 3 |- ( ( ph /\ x = A ) -> x = B )
5		csbied2.3	. . 3 |- ( ( ph /\ x = B ) -> C = D )
6	4, 5	syldan 280	. 2 |- ( ( ph /\ x = A ) -> C = D )
7	1, 6	csbied 3091	1 |- ( ph -> [_ A / x ]_ C = D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1343   e. wcel 2136   [_csb 3045

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-sbc 2952  df-csb 3046

This theorem is referenced by: (None)