Theorem psseq12d 3663

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 3661	. 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3		psseq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	psseq2d 3662	. 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))
5	2, 4	bitrd 267	1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ⊊ wpss 3541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-ne 2782  df-in 3547  df-ss 3554  df-pss 3556

This theorem is referenced by:  fin23lem32  9049  fin23lem34  9051  fin23lem35  9052  fin23lem41  9057  isf32lem5  9062  isf32lem6  9063  isf32lem11  9068  compssiso  9079  canthp1lem2  9354  chnle  27757