Theorem sbc2ie 3505

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 1843	. . 3 |- F/ x ps
4		nfv 1843	. . 3 |- F/ y ps
5	2	nfth 1727	. . 3 |- F/ x B e. _V
6		sbc2ie.3	. . 3 |- ( ( x = A /\ y = B ) -> ( ph <-> ps ) )
7	3, 4, 5, 6	sbc2iegf 3504	. 2 |- ( ( A e. _V /\ B e. _V ) -> ( [. A / x ]. [. B / y ]. ph <-> ps ) )
8	1, 2, 7	mp2an 708	1 |- ( [. A / x ]. [. B / y ]. ph <-> ps )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbc3ie  3507  brfi1uzind  13280  opfi1uzind  13283  brfi1uzindOLD  13286  opfi1uzindOLD  13289  wrd2ind  13477  isprs  16930  isdrs  16934  istos  17035  issrg  18507  isslmd  29755  rexrabdioph  37358  rmydioph  37581  rmxdioph  37583  expdiophlem2  37589