Theorem 3sstr3g 3965

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 1-Oct-2000.)

Hypotheses

Ref	Expression
3sstr3g.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr3g.2	⊢ 𝐴 = 𝐶
3sstr3g.3	⊢ 𝐵 = 𝐷

Assertion

Ref	Expression
3sstr3g	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr3g

Step	Hyp	Ref	Expression
1		3sstr3g.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr3g.2	. . 3 ⊢ 𝐴 = 𝐶
3		3sstr3g.3	. . 3 ⊢ 𝐵 = 𝐷
4	2, 3	sseq12i 3951	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)
5	1, 4	sylib 217	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-in 3894  df-ss 3904

This theorem is referenced by:  complss  4081  uniintsn  4918  fpwwe2lem12  10398  hmeocls  22919  hmeontr  22920  usgrumgruspgr  27550  chsscon3i  29823  pjss1coi  30525  mdslmd2i  30692  satffunlem2lem2  33368  ssbnd  35946  bnd2lem  35949  trclubgNEW  41226  nzss  41935