Theorem psseq1d 4118

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

Hypothesis

Ref	Expression
psseq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
psseq1d	⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))

Proof of Theorem psseq1d

Step	Hyp	Ref	Expression
1		psseq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		psseq1 4113	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ⊊ wpss 3977

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-ne 2947  df-ss 3993  df-pss 3996

This theorem is referenced by:  psseq12d  4120  fin23lem32  10413  fin23lem35  10416  compssiso  10443  mrieqv2d  17697  mrissmrcd  17698  pgpfac1lem5  20123  islbs3  21180  chpsscon2  31537