Theorem psseq1d 4061

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 4056	. 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 1540   ⊊ wpss 3918

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-cleq 2722  df-ne 2927  df-ss 3934  df-pss 3937

This theorem is referenced by:  psseq12d  4063  fin23lem32  10304  fin23lem35  10307  compssiso  10334  mrieqv2d  17607  mrissmrcd  17608  pgpfac1lem5  20018  islbs3  21072  chpsscon2  31441