Theorem csbied2 3052

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 2195	. . 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 3051	1 |- ( ph -> [_ A / x ]_ C = D )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   [_csb 3007

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-sbc 2914  df-csb 3008

This theorem is referenced by: (None)