Theorem psseq1i 4068

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 4066	. 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶))
3	1, 2	ax-mp 5	1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)

Syntax hints:   ↔ wb 208   = wceq 1537   ⊊ wpss 3939

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-ne 3019  df-in 3945  df-ss 3954  df-pss 3956

This theorem is referenced by:  psseq12i  4070