Theorem psseq12d 3926

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 3924	. 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3		psseq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	psseq2d 3925	. 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))
5	2, 4	bitrd 271	1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))

Syntax hints:   → wi 4   ↔ wb 198   = wceq 1658   ⊊ wpss 3798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-ext 2802

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2811  df-cleq 2817  df-clel 2820  df-ne 2999  df-in 3804  df-ss 3811  df-pss 3813

This theorem is referenced by:  fin23lem32  9480  fin23lem34  9482  fin23lem35  9483  fin23lem41  9488  isf32lem5  9493  isf32lem6  9494  isf32lem11  9499  compssiso  9510  canthp1lem2  9789  chnle  28927