Theorem 3sstr4g 3281

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 3266	. 2 ⊢ (𝐶 ⊆ 𝐷 ↔ 𝐴 ⊆ 𝐵)
5	1, 4	sylibr 134	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐷)

Syntax hints:   → wi 4   = wceq 1398   ⊆ wss 3211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-11 1555  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-in 3217  df-ss 3224

This theorem is referenced by:  rabss2  3321  unss2  3390  sslin  3447  ssopab2  4394  xpss12  4857  coss1  4910  coss2  4911  cnvss  4928  rnss  4987  ssres  5064  ssres2  5065  imass1  5137  imass2  5138  imadif  5436  imain  5438  ssoprab2  6109  suppssov1  6263  ressuppss  6454  tposss  6477  ss2ixp  6946  isumsplit  12177  isumrpcl  12180