Theorem sbcbig 3024

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 2980	. 2 |- ( y = A -> ( [ y / x ] ( ph <-> ps ) <-> [. A / x ]. ( ph <-> ps ) ) )
2		dfsbcq2 2980	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2980	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	bibi12d 235	. 2 |- ( y = A -> ( ( [ y / x ] ph <-> [ y / x ] ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )
5		sbbi 1971	. 2 |- ( [ y / x ] ( ph <-> ps ) <-> ( [ y / x ] ph <-> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2813	1 |- ( A e. V -> ( [. A / x ]. ( ph <-> ps ) <-> ( [. A / x ]. ph <-> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1364   [wsb 1773   e. wcel 2160   [.wsbc 2977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-sbc 2978

This theorem is referenced by:  sbcbi1  3027  sbcabel  3059