Theorem sbccsb 3849

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 2454	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3391	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2 3831	. 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 1767   {cab 2452   [.wsbc 3331   [_csb 3435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-in 3483  df-ss 3490  df-nul 3786

This theorem is referenced by: (None)