Theorem sseq12i 3593

Description: An equality inference for the subclass relationship. (Contributed by NM, 31-May-1999.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
sseq1i.1	⊢ 𝐴 = 𝐵
sseq12i.2	⊢ 𝐶 = 𝐷

Assertion

Ref	Expression
sseq12i	⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)

Proof of Theorem sseq12i

Step	Hyp	Ref	Expression
1		sseq1i.1	. 2 ⊢ 𝐴 = 𝐵
2		sseq12i.2	. 2 ⊢ 𝐶 = 𝐷
3		sseq12 3590	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
4	1, 2, 3	mp2an 703	1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)

Syntax hints:   ↔ wb 194   = wceq 1474   ⊆ wss 3539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589

This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-in 3546  df-ss 3553

This theorem is referenced by:  3sstr3i  3605  3sstr4i  3606  3sstr3g  3607  3sstr4g  3608  ss2rab  3640  rabsssn  4161  pjordi  28222  mdsldmd1i  28380  iuninc  28567  cvmlift2lem12  30356  brtrclfv2  36841  nzss  37341  hoidmvle  39294  ovolval5lem3  39348  issubgr  40497  fldhmsubc  41878  fldhmsubcALTV  41897