Theorem psseq2i 3804

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
psseq2i	⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)

Proof of Theorem psseq2i

Step	Hyp	Ref	Expression
1		psseq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		psseq2 3802	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
3	1, 2	ax-mp 5	1 ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)

Syntax hints:   ↔ wb 196   = wceq 1596   ⊊ wpss 3681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-10 2132  ax-11 2147  ax-12 2160  ax-13 2355  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1599  df-ex 1818  df-nf 1823  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-ne 2897  df-in 3687  df-ss 3694  df-pss 3696

This theorem is referenced by:  psseq12i  3805  disjpss  4136  infeq5i  8646