Theorem sbcrex 2893

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 2219	. 2 |- F/_ y A
2		sbcrext 2891	. 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 2206   E.wrex 2349   [.wsbc 2815

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 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816

This theorem is referenced by:  ac6sfi  6379  rexfiuz  9875