Theorem sbcbig 2921

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

Assertion

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

Proof of Theorem sbcbig

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2879	. 2 |- ( y = A -> ( [ y / x ] ( ph <-> ps ) <-> [. A / x ]. ( ph <-> ps ) ) )
2		dfsbcq2 2879	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2879	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	bibi12d 234	. 2 |- ( y = A -> ( ( [ y / x ] ph <-> [ y / x ] ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
5		sbbi 1906	. 2 |- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2716	1 |- ( A e. V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1312   e. wcel 1461   [wsb 1716   [.wsbc 2876

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 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-v 2657  df-sbc 2877

This theorem is referenced by:  sbcbi1  2924  sbcabel  2956