Theorem sbcbidv 3057

Description: Formula-building deduction 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 1551	. 2 |- F/ x ph
2		sbcbidv.1	. 2 |- ( ph -> ( ps <-> ch ) )
3	1, 2	sbcbid 3056	1 |- ( ph -> ( [. A / x ]. ps <-> [. A / x ]. ch ) )

Syntax hints:   -> wi 4   <-> wb 105   [.wsbc 2998

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-sbc 2999

This theorem is referenced by:  sbcbii  3058  csbcomg  3116  opelopabsb  4306  opelopabgf  4316  opelopabf  4321  sbcfng  5423  sbcfg  5424  uchoice  6223  f1od2  6321  islmod  14053