Theorem sbcrex 2904

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 2223	. 2 |- F/_ y A
2		sbcrext 2902	. 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 2210   E.wrex 2354   [.wsbc 2826

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2614  df-sbc 2827

This theorem is referenced by:  ac6sfi  6544  rexfiuz  10249