Theorem eqsstrrd 3229

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

Hypotheses

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

Assertion

Ref	Expression
eqsstrrd	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem eqsstrrd

Step	Hyp	Ref	Expression
1		eqsstrrd.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2210	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqsstrrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	eqsstrd 3228	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1372   ⊆ wss 3165

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-11 1528  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-in 3171  df-ss 3178

This theorem is referenced by:  ssxpbm  5117  ssxp1  5118  ssxp2  5119  suppssof1  6175  tfrlemiubacc  6415  tfr1onlemubacc  6431  tfrcllemubacc  6444  oaword1  6556  phplem4dom  6958  fisseneq  7030  nnnninfeq2  7230  archnqq  7529  hashdmprop2dom  10987  imasaddfnlemg  13117  resmhm2  13291  ringidss  13762  subrg1  13964  subrgdvds  13968  subrguss  13969  subrginv  13970  islss3  14112  lspsnneg  14153  epttop  14533  metequiv2  14939  limccnpcntop  15118  limccnp2lem  15119  limccnp2cntop  15120  nnsf  15904