Theorem sbc7 2880

Description: An equivalence for class substitution in the spirit of df-clab 2082. 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 2875	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
2		sbc5 2877	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )
3	1, 2	bitr3i 185	1 |- ( [. A / x ]. ph <-> E. y ( y = A /\ [. y / x ]. ph ) )

Syntax hints:   /\ wa 103   <-> wb 104   = wceq 1296   E.wex 1433   [.wsbc 2854

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-v 2635  df-sbc 2855

This theorem is referenced by: (None)