Theorem sbc7 2790

Description: An equivalence for class substitution in the spirit of df-clab 2027. 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 2785	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
2		sbc5 2787	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )
3	1, 2	bitr3i 175	1 |- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )

Syntax hints:   /\ wa 97   <-> wb 98   = wceq 1243   E.wex 1381   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765

This theorem is referenced by: (None)