Theorem psseq1i 4019

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 4017	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ⊊ wpss 4013

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-cleq 2752  df-ne 2956  df-ss 3896  df-pss 4014

This theorem is used by:  psseq12i  4021