Theorem sbcbig 3480

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 3438	. 2 |- ( y = A -> ( [ y / x ] ( ph <-> ps ) <-> [. A / x ]. ( ph <-> ps ) ) )
2		dfsbcq2 3438	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 3438	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	bibi12d 335	. 2 |- ( y = A -> ( ( [ y / x ] ph <-> [ y / x ] ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
5		sbbi 2401	. 2 |- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 3267	1 |- ( A e. V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   [wsb 1880   e. wcel 1990   [.wsbc 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  sbcbi1  3483  sbcabel  3517  bnj89  30787  bj-sbeq  32896  bj-sbceqgALT  32897  sbcbi  38749  sbc3orgVD  39086  sbcbiVD  39112