Theorem sbc2ie 3123

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
sbc2ie.1	|- A e. _V
sbc2ie.2	|- B e. _V
sbc2ie.3	|- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )

Assertion

Ref	Expression
sbc2ie	|- ( [. A / x ]. [. B / y ]. ph <-> ps )

Distinct variable groups:   x, y, A   y, B   ps, x, y

Allowed substitution hints:   ph( x, y)   B( x)

Proof of Theorem sbc2ie

Step	Hyp	Ref	Expression
1		sbc2ie.1	. 2 |- A e. _V
2		sbc2ie.2	. 2 |- B e. _V
3		nfv 1581	. . 3 |- F/ x ps
4		nfv 1581	. . 3 |- F/ y ps
5	2	nfth 1517	. . 3 |- F/ x B e. _V
6		sbc2ie.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	3, 4, 5, 6	sbc2iegf 3122	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
8	1, 2, 7	mp2an 430	1 |- ( [. A / x ]. [. B / y ]. ph <-> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1402   e. wcel 2209   _Vcvv 2821   [.wsbc 3051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052

This theorem is referenced by:  sbc3ie  3125  wrd2ind  11473