Theorem sbc7 2981

Description: An equivalence for class substitution in the spirit of df-clab 2157. 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 2976	. 2 |- ( [. A / y ]. [. y / x ]. ph <-> [. A / x ]. ph )
2		sbc5 2978	. 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 1348   E.wex 1485   [.wsbc 2955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956

This theorem is referenced by: (None)