Theorem sseq12i 3975

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 3972	. 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷))
4	1, 2, 3	mp2an 704	1 ⊢ (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷)

Syntax hints:   ↔ wb 209   = wceq 1567   ⊆ wss 3913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-cleq 2761  df-ss 3930

This theorem is referenced by:  ss2rab  4031  rabsssn  4636  fldhmsubc  20862  issubgr  29558  pjordi  32462  mdsldmd1i  32620  rabsspr  32784  rabsstp  32785  iuninc  32842  cvmlift2lem12  35701  brtrclfv2  44338  nzss  44912  hoidmvle  47199  fldhmsubcALTV  48980