Theorem cleq2lem 37914

Description: Equality implies bijection. (Contributed by RP, 24-Jul-2020.)

Hypothesis

Ref	Expression
cleq2lem.b	|- ( A = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
cleq2lem	|- ( A = B -> ( ( R C_ A /\ ph ) <-> ( R C_ B /\ ps ) ) )

Proof of Theorem cleq2lem

Step	Hyp	Ref	Expression
1		sseq2 3627	. 2 |- ( A = B -> ( R C_ A <-> R C_ B ) )
2		cleq2lem.b	. 2 |- ( A = B -> ( ph <-> ps ) )
3	1, 2	anbi12d 747	1 |- ( A = B -> ( ( R C_ A /\ ph ) <-> ( R C_ B /\ ps ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574

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-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-in 3581  df-ss 3588

This theorem is referenced by:  cbvcllem  37915  clublem  37917  rclexi  37922  rtrclex  37924  rtrclexi  37928  clrellem  37929  clcnvlem  37930  trcleq2lemRP  37937  dfrcl2  37966  brtrclfv2  38019  clsk1indlem1  38343