Theorem 3sstr3g 3967

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.2	. . 3 ⊢ 𝐴 = 𝐶
2		3sstr3g.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3	1, 2	eqsstrrid 3954	. 2 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)
4		3sstr3g.3	. 2 ⊢ 𝐵 = 𝐷
5	3, 4	sseqtrdi 3955	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1547   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787  df-cleq 2731  df-ss 3900

This theorem is referenced by:  complss  4081  uniintsn  4915  fpwwe2lem12  10556  hmeocls  23751  hmeontr  23752  usgrumgruspgr  29269  chsscon3i  31550  pjss1coi  32252  mdslmd2i  32419  satffunlem2lem2  35634  ssbnd  38155  bnd2lem  38158  trclubgNEW  44062  nzss  44761