Theorem 3sstr4g 2986

Description: Substitution of equality into both sides of a subclass relationship. (Contributed by NM, 16-Aug-1994.) (Proof shortened by Eric Schmidt, 26-Jan-2007.)

Hypotheses

Ref	Expression
3sstr4g.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3sstr4g.2	⊢ 𝐶 = 𝐴
3sstr4g.3	⊢ 𝐷 = 𝐵

Assertion

Ref	Expression
3sstr4g	⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Proof of Theorem 3sstr4g

Step	Hyp	Ref	Expression
1		3sstr4g.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		3sstr4g.2	. . 3 ⊢ 𝐶 = 𝐴
3		3sstr4g.3	. . 3 ⊢ 𝐷 = 𝐵
4	2, 3	sseq12i 2971	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	sylibr 137	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  rabss2  3023  unss2  3114  sslin  3163  ssopab2  4012  xpss12  4445  coss1  4491  coss2  4492  cnvss  4508  rnss  4564  ssres  4637  ssres2  4638  imass1  4700  imass2  4701  imadif  4979  imain  4981  ssoprab2  5561  suppssfv  5708  suppssov1  5709  tposss  5861