Theorem sbcrex 2918

Description: Interchange class substitution and restricted existential quantifier. (Contributed by NM, 15-Nov-2005.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbcrex	|- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph )

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

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

Proof of Theorem sbcrex

Step	Hyp	Ref	Expression
1		nfcv 2228	. 2 |- F/_ y A
2		sbcrext 2916	. 2 |- ( F/_ y A -> ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph ) )
3	1, 2	ax-mp 7	1 |- ( [. A / x ]. E. y e. B ph <-> E. y e. B [. A / x ]. ph )

Syntax hints:   <-> wb 103   F/_wnfc 2215   E.wrex 2360   [.wsbc 2840

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-sbc 2841

This theorem is referenced by:  ac6sfi  6612  rexfiuz  10418