Theorem sbc7 3024

Description: An equivalence for class substitution in the spirit of df-clab 2191. Note that x and A don't have to be distinct. (Contributed by NM, 18-Nov-2008.) (Revised by Mario Carneiro, 13-Oct-2016.)

Assertion

Ref	Expression
sbc7	|- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )

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

Allowed substitution hints:   ph( x)   A( x)

Proof of Theorem sbc7

Step	Hyp	Ref	Expression
1		sbcco 3019	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
2		sbc5 3021	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )
3	1, 2	bitr3i 186	1 |- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )

Syntax hints:   /\ wa 104   <-> wb 105   = wceq 1372   E.wex 1514   [.wsbc 2997

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-sbc 2998

This theorem is referenced by: (None)