Theorem psseq2d 4037

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 4032	. 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))
3	1, 2	syl 17	1 ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1542   ⊊ wpss 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ne 2934  df-ss 3907  df-pss 3910

This theorem is referenced by:  psseq12d  4038  php3  9136  inf3lem5  9544  infeq5i  9548  ackbij1lem15  10146  fin4en1  10222  chpsscon1  31590  chnle  31600  atcvatlem  32471  atcvati  32472  lsatcvat  39510