Theorem sseq12i 3943

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 3940	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
4	1, 2, 3	mp2an 705	1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-cleq 2755  df-ss 3901

This theorem is used by:  ss2rab  3997  rabsssn  4630  fldhmsubc  21093  issubgr  29909  pjordi  32832  mdsldmd1i  32990  rabsspr  33196  rabsstp  33197  iuninc  33253  cvmlift2lem12  36287  brtrclfv2  44988  nzss  45562  cocanss1  46185  hoidmvle  47883  fldhmsubcALTV  49703