Theorem sbcbid 1986

Description: Formula-building deduction rule for class substitution.

Hypotheses

Ref	Expression
sbcbid.1	|- (ph -> A.xph)
sbcbid.2	|- (ph -> (ps <-> ch))

Assertion

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

Proof of Theorem sbcbid

Step	Hyp	Ref	Expression
1		a4sbc 1952	. . . 4 |- (A e. B -> (A.x(ps <-> ch) -> [A / x](ps <-> ch)))
2		sbcbid.1	. . . . 5 |- (ph -> A.xph)
3		sbcbid.2	. . . . 5 |- (ph -> (ps <-> ch))
4	2, 3	19.21ai 1002	. . . 4 |- (ph -> A.x(ps <-> ch))
5	1, 4	syl5 21	. . 3 |- (A e. B -> (ph -> [A / x](ps <-> ch)))
6		sbcbidig 1983	. . 3 |- (A e. B -> ([A / x](ps <-> ch) <-> ([A / x]ps <-> [A / x]ch)))
7	5, 6	sylibd 202	. 2 |- (A e. B -> (ph -> ([A / x]ps <-> [A / x]ch)))
8	7	impcom 351	1 |- ((ph /\ A e. B) -> ([A / x]ps <-> [A / x]ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 958   e. wcel 962  [wsbc 1174

This theorem is referenced by:  sbcbidv 1987  hbsbcgd 1994  sbcnestg 2049

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 966  ax-gen 967  ax-8 968  ax-10 970  ax-12 972  ax-17 975  ax-4 977  ax-5o 979  ax-6o 982  ax-9o 1127  ax-10o 1144  ax-11o 1222  ax-ext 1464

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985  df-sb 1176  df-clab 1470  df-cleq 1475  df-clel 1478  df-v 1819  df-sbc 1949

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