Theorem csbie2g 3182

Description: Conversion of implicit substitution to explicit class substitution. This version of sbcie 3080 avoids a disjointness condition on x, A by substituting twice. (Contributed by Mario Carneiro, 11-Nov-2016.)

Hypotheses

Ref	Expression
csbie2g.1	|- (x = y -> B = C)
csbie2g.2	|- (y = A -> C = D)

Assertion

Ref	Expression
csbie2g	|- (A e. V -> [_A / x]_B = D)

Distinct variable groups:   x,y   y,A   y,B   x,C   y,D

Allowed substitution hints:   A(x)   B(x)   C(y)   D(x)   V(x,y)

Proof of Theorem csbie2g

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-csb 3137	. 2 |- [_A / x]_B = {z | [.A / x ].z e. B}
2		csbie2g.1	. . . . 5 |- (x = y -> B = C)
3	2	eleq2d 2420	. . . 4 |- (x = y -> (z e. B <-> z e. C))
4		csbie2g.2	. . . . 5 |- (y = A -> C = D)
5	4	eleq2d 2420	. . . 4 |- (y = A -> (z e. C <-> z e. D))
6	3, 5	sbcie2g 3079	. . 3 |- (A e. V -> ( [.A / x ].z e. B <-> z e. D))
7	6	abbi1dv 2469	. 2 |- (A e. V -> {z | [.A / x ].z e. B} = D)
8	1, 7	syl5eq 2397	1 |- (A e. V -> [_A / x]_B = D)

Syntax hints:   -> wi 4   = wceq 1642   e. wcel 1710  {cab 2339   [.wsbc 3046  [_csb 3136

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-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-sbc 3047  df-csb 3137

This theorem is referenced by: (None)