Theorem sbcang 1967

Description: Distribution of class substitution over conjunction.

Assertion

Ref	Expression
sbcang	|- (A e. B -> ([A / x](ph /\ ps) <-> ([A / x]ph /\ [A / x]ps)))

Proof of Theorem sbcang

Step	Hyp	Ref	Expression
1		dfsbcq 1939	. 2 |- (y = A -> ([y / x](ph /\ ps) <-> [A / x](ph /\ ps)))
2		dfsbcq 1939	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
3		dfsbcq 1939	. . 3 |- (y = A -> ([y / x]ps <-> [A / x]ps))
4	2, 3	anbi12d 627	. 2 |- (y = A -> (([y / x]ph /\ [y / x]ps) <-> ([A / x]ph /\ [A / x]ps)))
5		sban 1235	. 2 |- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
6	1, 4, 5	vtoclbg 1844	1 |- (A e. B -> ([A / x](ph /\ ps) <-> ([A / x]ph /\ [A / x]ps)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 954   e. wcel 956  [wsbc 1168

This theorem is referenced by:  sbc3ang 1975  sbcabel 1992  sbcel12g 2007  intab 2555  csbopabg 2673  dfoprab5 4105  foprab2 4109  fsumcnlem 7939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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