Theorem sbccsb 3856

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 2444	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3387	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2 3839	. 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 1819   {cab 2442   [.wsbc 3327   [_csb 3430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-in 3478  df-ss 3485  df-nul 3794

This theorem is referenced by: (None)