Theorem eqsstr3d 3084

Description: Substitution of equality into a subclass relationship. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
eqsstr3d.1	⊢ (𝜑 → 𝐵 = 𝐴)
eqsstr3d.2	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
eqsstr3d	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem eqsstr3d

Step	Hyp	Ref	Expression
1		eqsstr3d.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2105	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqsstr3d.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	eqsstrd 3083	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1299   ⊆ wss 3021

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-11 1452  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-in 3027  df-ss 3034

This theorem is referenced by:  ssxpbm  4910  ssxp1  4911  ssxp2  4912  suppssof1  5930  tfrlemiubacc  6157  tfr1onlemubacc  6173  tfrcllemubacc  6186  oaword1  6297  phplem4dom  6685  fisseneq  6749  archnqq  7126  epttop  12041  metequiv2  12424  limccnpcntop  12520  nnsf  12783