Theorem psseq12d 4094

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

Hypotheses

Ref	Expression
psseq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)
psseq12d.2	⊢ (𝜑 → 𝐶 = 𝐷)

Assertion

Ref	Expression
psseq12d	⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))

Proof of Theorem psseq12d

Step	Hyp	Ref	Expression
1		psseq1d.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2	1	psseq1d 4092	. 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3		psseq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	psseq2d 4093	. 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))
5	2, 4	bitrd 278	1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ⊊ wpss 3950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2938  df-v 3475  df-in 3956  df-ss 3966  df-pss 3968

This theorem is referenced by:  fin23lem32  10375  fin23lem34  10377  fin23lem35  10378  fin23lem41  10383  isf32lem5  10388  isf32lem6  10389  isf32lem11  10394  compssiso  10405  chnle  31344