Theorem eqsstr3d 3034

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 2086	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqsstr3d.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	eqsstrd 3033	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1284   ⊆ wss 2973

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-11 1437  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-in 2979  df-ss 2986

This theorem is referenced by:  ssxpbm  4776  ssxp1  4777  ssxp2  4778  suppssof1  5748  tfrlemiubacc  5967  oaword1  6073  phplem4dom  6348  archnqq  6607