Theorem sbcimg 1973

Description: Distribution of class substitution over implication.

Assertion

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

Proof of Theorem sbcimg

Step	Hyp	Ref	Expression
1		dfsbcq 1946	. 2 |- (y = A -> ([y / x](ph -> ps) <-> [A / x](ph -> ps)))
2		dfsbcq 1946	. . 3 |- (y = A -> ([y / x]ph <-> [A / x]ph))
3		dfsbcq 1946	. . 3 |- (y = A -> ([y / x]ps <-> [A / x]ps))
4	2, 3	imbi12d 628	. 2 |- (y = A -> (([y / x]ph -> [y / x]ps) <-> ([A / x]ph -> [A / x]ps)))
5		sbim 1236	. 2 |- ([y / x](ph -> ps) <-> ([y / x]ph -> [y / x]ps))
6	1, 4, 5	vtoclbg 1851	1 |- (A e. B -> ([A / x](ph -> ps) <-> ([A / x]ph -> [A / x]ps)))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   e. wcel 960  [wsbc 1172

This theorem is referenced by:  sbcth2 1985  sbc19.20dv 1988  sbc19.21g 1990  ra4sbc 2000  intab 2564  reuuniss2 2897

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-v 1815  df-sbc 1945

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