Theorem psseq12d 2145

Description: An equality deduction for the proper subclass relationship.

Hypotheses

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

Assertion

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

Proof of Theorem psseq12d

Step	Hyp	Ref	Expression
1		psseq1d.1	. . 3 |- (ph -> A = B)
2	1	psseq1d 2143	. 2 |- (ph -> (A (. C <-> B (. C))
3		psseq12d.2	. . 3 |- (ph -> C = D)
4	3	psseq2d 2144	. 2 |- (ph -> (B (. C <-> B (. D))
5	2, 4	bitrd 530	1 |- (ph -> (A (. C <-> B (. D))

Syntax hints:   -> wi 3   <-> wb 146   = wceq 958   (. wpss 2051

This theorem is referenced by:  chnlet 9432

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-in 2054  df-ss 2056  df-pss 2058

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