Theorem csbied2 3180

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 2407	. . 3 |- ((ph /\ x = A) -> x = B)
5		csbied2.3	. . 3 |- ((ph /\ x = B) -> C = D)
6	4, 5	syldan 456	. 2 |- ((ph /\ x = A) -> C = D)
7	1, 6	csbied 3179	1 |- (ph -> [_A / x]_C = D)

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   e. wcel 1710  [_csb 3137

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-sbc 3048  df-csb 3138

This theorem is referenced by: (None)