Theorem psseq12d 3038

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 3036	. 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3		psseq12d.2	. . 3 ⊢ (𝜑 → 𝐶 = 𝐷)
4	3	psseq2d 3037	. 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))
5	2, 4	bitrd 177	1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷))

Syntax hints:   → wi 4   ↔ wb 98   = wceq 1243   ⊊ wpss 2918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-ne 2206  df-in 2924  df-ss 2931  df-pss 2933

This theorem is referenced by: (None)