Theorem psseq1i 4115

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 4113	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)

Syntax hints:   ↔ 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:  psseq12i  4117