Theorem sbcimg 2918

Description: Distribution of class substitution over implication. (Contributed by NM, 16-Jan-2004.)

Assertion

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

Proof of Theorem sbcimg

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfsbcq2 2881	. 2 |- ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
2		dfsbcq2 2881	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 2881	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	imbi12d 233	. 2 |- ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
5		sbim 1902	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2718	1 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1314   e. wcel 1463   [wsb 1718   [.wsbc 2878

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 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-sbc 2879

This theorem is referenced by:  sbcim1  2925  sbceqal  2932  sbc19.21g  2945  sbcssg  3438  iota4an  5065  sbcfung  5105  riotass2  5710