Theorem trcleq12lem 13732

Description: Equality implies bijection. (Contributed by RP, 9-May-2020.)

Assertion

Ref	Expression
trcleq12lem	|- ( ( R = S /\ A = B ) -> ( ( R C_ A /\ ( A o. A ) C_ A ) <-> ( S C_ B /\ ( B o. B ) C_ B ) ) )

Proof of Theorem trcleq12lem

Step	Hyp	Ref	Expression
1		cleq1lem 13721	. 2 |- ( R = S -> ( ( R C_ A /\ ( A o. A ) C_ A ) <-> ( S C_ A /\ ( A o. A ) C_ A ) ) )
2		trcleq2lem 13730	. 2 |- ( A = B -> ( ( S C_ A /\ ( A o. A ) C_ A ) <-> ( S C_ B /\ ( B o. B ) C_ B ) ) )
3	1, 2	sylan9bb 736	1 |- ( ( R = S /\ A = B ) -> ( ( R C_ A /\ ( A o. A ) C_ A ) <-> ( S C_ B /\ ( B o. B ) C_ B ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   C_ wss 3574   o. ccom 5118

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-nfc 2753  df-in 3581  df-ss 3588  df-br 4654  df-opab 4713  df-co 5123

This theorem is referenced by: (None)