Theorem sbcimg 3093

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 3054	. 2 |- ( y = A -> ( [ y / x ] ( ph -> ps ) <-> [. A / x ]. ( ph -> ps ) ) )
2		dfsbcq2 3054	. . 3 |- ( y = A -> ( [ y / x ] ph <-> [. A / x ]. ph ) )
3		dfsbcq2 3054	. . 3 |- ( y = A -> ( [ y / x ] ps <-> [. A / x ]. ps ) )
4	2, 3	imbi12d 234	. 2 |- ( y = A -> ( ( [ y / x ] ph -> [ y / x ] ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )
5		sbim 2013	. 2 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
6	1, 4, 5	vtoclbg 2884	1 |- ( A e. V -> ( [. A / x ]. ( ph -> ps ) <-> ( [. A / x ]. ph -> [. A / x ]. ps ) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   [wsb 1815   e. wcel 2209   [.wsbc 3051

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-sbc 3052

This theorem is referenced by:  sbcim1  3100  sbceqal  3107  sbc19.21g  3120  sbcssg  3633  iota4an  5353  sbcfung  5396  riotass2  6057