Theorem sbcbidv 2817

Description: Formula-building deduction rule for class substitution. (Contributed by NM, 29-Dec-2014.)

Hypothesis

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

Assertion

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

Distinct variable group:   ph, x

Allowed substitution hints:   ps( x)   ch( x)   A( x)

Proof of Theorem sbcbidv

Step	Hyp	Ref	Expression
1		nfv 1421	. 2 |- F/ x ph
2		sbcbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	sbcbid 2816	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 98   [.wsbc 2764

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-sbc 2765

This theorem is referenced by:  sbcbii  2818  csbcomg  2873  opelopabsb  3997  opelopabf  4011  sbcfng  5044  sbcfg  5045