Theorem sbcbidig 1963

Description: Distribution of class substitution over biconditional. (Contributed by Raph Levien, 10-Apr-2004.)

Assertion

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

Proof of Theorem sbcbidig

Step	Hyp	Ref	Expression
1		dfsbcq 1933	. 2 |- (y = A -> ([y / x](ph <-> ps) <-> [A / x](ph <-> ps)))
2		dfsbcq 1933	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
3		dfsbcq 1933	. . 3 |- (y = A -> ([y / x]ps <-> [A / x]ps))
4	2, 3	bibi12d 627	. 2 |- (y = A -> (([y / x]ph <-> [y / x]ps) <-> ([A / x]ph <-> [A / x]ps)))
5		sbbi 1234	. 2 |- ([y / x](ph <-> ps) <-> ([y / x]ph <-> [y / x]ps))
6	1, 4, 5	vtoclbg 1839	1 |- (A e. B -> ([A / x](ph <-> ps) <-> ([A / x]ph <-> [A / x]ps)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 953   e. wcel 955  [wsbc 1166

This theorem is referenced by:  sbcbid 1966  sbcabel 1986  sbcel12g 2001  sbceqdig 2002

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-v 1803  df-sbc 1932

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