Theorem sbccsb 4004

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 2610	. . 3 |- ( y e. { y | ph } <-> ph )
2	1	sbcbii 3491	. 2 |- ( [. A / x ]. y e. { y | ph } <-> [. A / x ]. ph )
3		sbcel2 3989	. 2 |- ( [. A / x ]. y e. { y | ph } <-> y e. [_ A / x ]_ { y | ph } )
4	2, 3	bitr3i 266	1 |- ( [. A / x ]. ph <-> y e. [_ A / x ]_ { y | ph } )

Syntax hints:   <-> wb 196   e. wcel 1990   {cab 2608   [.wsbc 3435   [_csb 3533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-nul 3916

This theorem is referenced by: (None)