Theorem psseq2d 4051

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

Hypothesis

Ref	Expression
psseq1d.1	⊢ (𝜑 → 𝐴 = 𝐵)

Assertion

Ref	Expression
psseq2d	⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))

Proof of Theorem psseq2d

Step	Hyp	Ref	Expression
1		psseq1d.1	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
2		psseq2 4046	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
3	1, 2	syl 18	1 ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1570   ⊊ wpss 3907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1810  df-cleq 2755  df-ne 2959  df-ss 3923  df-pss 3926

This theorem is referenced by:  psseq12d  4052  php3  9194  inf3lem5  9602  infeq5i  9606  ackbij1lem15  10217  fin4en1  10294  chpsscon1  31834  chnle  31844  atcvatlem  32715  atcvati  32716  lsatcvat  39802