Theorem psseq12i 3361

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

Hypotheses

Ref	Expression
psseq1i.1	⊢ A = B
psseq12i.2	⊢ C = D

Assertion

Ref	Expression
psseq12i	⊢ (A ⊊ C ↔ B ⊊ D)

Proof of Theorem psseq12i

Step	Hyp	Ref	Expression
1		psseq1i.1	. . 3 ⊢ A = B
2	1	psseq1i 3359	. 2 ⊢ (A ⊊ C ↔ B ⊊ C)
3		psseq12i.2	. . 3 ⊢ C = D
4	3	psseq2i 3360	. 2 ⊢ (B ⊊ C ↔ B ⊊ D)
5	2, 4	bitri 240	1 ⊢ (A ⊊ C ↔ B ⊊ D)

Syntax hints:   ↔ wb 176   = wceq 1642   ⊊ wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by: (None)