Theorem psseq1i 4028

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

Hypothesis

Ref	Expression
psseq1i.1	⊢ 𝐴 = 𝐵

Assertion

Ref	Expression
psseq1i	⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)

Proof of Theorem psseq1i

Step	Hyp	Ref	Expression
1		psseq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		psseq1 4026	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)

Syntax hints:   ↔ wb 205   = wceq 1541   ⊊ wpss 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1544  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ne 2945  df-v 3432  df-in 3898  df-ss 3908  df-pss 3910

This theorem is referenced by:  psseq12i  4030