Theorem sbcbidv 1973

Description: Formula-building deduction rule for class substitution. (The proof was shortened by Eric Schmidt, 17-Jan-2007.)

Hypothesis

Ref	Expression
sbcbidv.1	|- (ph -> (ps <-> ch))

Assertion

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

Distinct variable group:   ph,x

Proof of Theorem sbcbidv

Step	Hyp	Ref	Expression
1		ax-17 969	. 2 |- (ph -> A.xph)
2		sbcbidv.1	. 2 |- (ph -> (ps <-> ch))
3	1, 2	sbcbid 1972	1 |- ((ph /\ A e. B) -> ([A / x]ps <-> [A / x]ch))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 956  [wsbc 1168

This theorem is referenced by:  sbcbii 1974  sbccomglem 1984  sbccomg 1985  sbcel12g 2007  sbceqdig 2008  csbcomg 2013  isarep1 3569

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-v 1808  df-sbc 1938

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