Theorem 3sstr3g 3864

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 3850	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)
5	1, 4	sylib 210	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1601   ⊆ wss 3792

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-in 3799  df-ss 3806

This theorem is referenced by:  complss  3974  uniintsn  4747  fpwwe2lem13  9799  hmeocls  21980  hmeontr  21981  usgrumgruspgr  26529  chsscon3i  28892  pjss1coi  29594  mdslmd2i  29761  ssbnd  34213  bnd2lem  34216  trclubgNEW  38886  nzss  39476