Theorem sbcang 1971

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 1943	. 2 |- (y = A -> ([y / x](ph /\ ps) <-> [A / x](ph /\ ps)))
2		dfsbcq 1943	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
3		dfsbcq 1943	. . 3 |- (y = A -> ([y / x]ps <-> [A / x]ps))
4	2, 3	anbi12d 628	. 2 |- (y = A -> (([y / x]ph /\ [y / x]ps) <-> ([A / x]ph /\ [A / x]ps)))
5		sban 1237	. 2 |- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
6	1, 4, 5	vtoclbg 1848	1 |- (A e. B -> ([A / x](ph /\ ps) <-> ([A / x]ph /\ [A / x]ps)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  [wsbc 1170

This theorem is referenced by:  sbc3ang 1979  sbcabel 1996  sbcel12g 2011  intab 2560  csbopabg 2678  dfoprab5 4115  foprab2 4119  fsumcnlem 7989

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-sbc 1942

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