Theorem bj-cleq 32949

Description: Substitution property for certain classes. (Contributed by BJ, 2-Apr-2019.)

Assertion

Ref	Expression
bj-cleq	|- ( A = B -> { x | { x } e. ( A " C ) } = { x | { x } e. ( B " C ) } )

Distinct variable groups:   x, A   x, B   x, C

Proof of Theorem bj-cleq

Step	Hyp	Ref	Expression
1		imaeq1 5461	. . 3 |- ( A = B -> ( A " C ) = ( B " C ) )
2		eleq2 2690	. . . 4 |- ( ( A " C ) = ( B " C ) -> ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) )
3	2	alrimiv 1855	. . 3 |- ( ( A " C ) = ( B " C ) -> A. x ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) )
4	1, 3	syl 17	. 2 |- ( A = B -> A. x ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) )
5		abbi 2737	. 2 |- ( A. x ( { x } e. ( A " C ) <-> { x } e. ( B " C ) ) <-> { x | { x } e. ( A " C ) } = { x | { x } e. ( B " C ) } )
6	4, 5	sylib 208	1 |- ( A = B -> { x | { x } e. ( A " C ) } = { x | { x } e. ( B " C ) } )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   e. wcel 1990   {cab 2608   {csn 4177   "cima 5117

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-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by: (None)