Theorem sbcbid 3100

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

Hypotheses

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

Assertion

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

Proof of Theorem sbcbid

Step	Hyp	Ref	Expression
1		sbcbid.1	. . . 4 |- F/xph
2		sbcbid.2	. . . 4 |- (ph -> (ps <-> ch))
3	1, 2	abbid 2467	. . 3 |- (ph -> {x | ps} = {x | ch})
4	3	eleq2d 2420	. 2 |- (ph -> (A e. {x | ps} <-> A e. {x | ch}))
5		df-sbc 3048	. 2 |- ( [.A / x ].ps <-> A e. {x | ps})
6		df-sbc 3048	. 2 |- ( [.A / x ].ch <-> A e. {x | ch})
7	4, 5, 6	3bitr4g 279	1 |- (ph -> ( [.A / x ].ps <-> [.A / x ].ch))

Syntax hints:   -> wi 4   <-> wb 176   F/wnf 1544   e. wcel 1710  {cab 2339   [.wsbc 3047

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-sbc 3048

This theorem is referenced by:  sbcbidv  3101  csbeq2d  3161