Theorem sbccsb 3810

Description: Substitution into a wff expressed in terms of substitution into a class. (Contributed by NM, 15-Aug-2007.) (Revised by NM, 18-Aug-2018.)

Assertion

Ref	Expression
sbccsb	|- ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } )

Distinct variable group:   x, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem sbccsb

Step	Hyp	Ref	Expression
1		abid 2441	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3354	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2 3792	. 2 |- ( [. A / x ]. y e. { y | ph } <-> y e. [_ A / x ]_ { y | ph } )
4	2, 3	bitr3i 251	1 |- ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } )

Syntax hints:   <-> wb 184   e. wcel 1758   {cab 2439   [.wsbc 3294   [_csb 3396

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3747

This theorem is referenced by: (None)