Theorem 3sstr3g 4053

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 4039	. 2 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐷)
5	1, 4	sylib 218	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1537   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-ss 3993

This theorem is referenced by:  complss  4174  uniintsn  5009  fpwwe2lem12  10711  hmeocls  23797  hmeontr  23798  usgrumgruspgr  29217  chsscon3i  31493  pjss1coi  32195  mdslmd2i  32362  satffunlem2lem2  35374  ssbnd  37748  bnd2lem  37751  trclubgNEW  43580  nzss  44286