Theorem psseq12d 3701

Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.)

Hypotheses

Ref	Expression
psseq1d.1	|- ( ph -> A = B )
psseq12d.2	|- ( ph -> C = D )

Assertion

Ref	Expression
psseq12d	|- ( ph -> ( A C. C <-> B C. D ) )

Proof of Theorem psseq12d

Step	Hyp	Ref	Expression
1		psseq1d.1	. . 3 |- ( ph -> A = B )
2	1	psseq1d 3699	. 2 |- ( ph -> ( A C. C <-> B C. C ) )
3		psseq12d.2	. . 3 |- ( ph -> C = D )
4	3	psseq2d 3700	. 2 |- ( ph -> ( B C. C <-> B C. D ) )
5	2, 4	bitrd 268	1 |- ( ph -> ( A C. C <-> B C. D ) )

Syntax hints:   -> wi 4   <-> wb 196   = wceq 1483   C. wpss 3575

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-ne 2795  df-in 3581  df-ss 3588  df-pss 3590

This theorem is referenced by:  fin23lem32  9166  fin23lem34  9168  fin23lem35  9169  fin23lem41  9174  isf32lem5  9179  isf32lem6  9180  isf32lem11  9185  compssiso  9196  canthp1lem2  9475  chnle  28373