Theorem psseq12d 3843

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 3841	. 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3		psseq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	psseq2d 3842	. 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))
5	2, 4	bitrd 268	1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1632   ⊊ wpss 3716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-ne 2933  df-in 3722  df-ss 3729  df-pss 3731

This theorem is referenced by:  fin23lem32  9358  fin23lem34  9360  fin23lem35  9361  fin23lem41  9366  isf32lem5  9371  isf32lem6  9372  isf32lem11  9377  compssiso  9388  canthp1lem2  9667  chnle  28682