Theorem eqsstrrd 3234

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 2212	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		eqsstrrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	eqsstrd 3233	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1373   ⊆ wss 3170

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 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-11 1530  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-in 3176  df-ss 3183

This theorem is referenced by:  ssxpbm  5132  ssxp1  5133  ssxp2  5134  suppssof1  6194  tfrlemiubacc  6434  tfr1onlemubacc  6450  tfrcllemubacc  6463  oaword1  6575  phplem4dom  6979  fisseneq  7052  nnnninfeq2  7252  archnqq  7560  hashdmprop2dom  11021  imasaddfnlemg  13231  resmhm2  13405  ringidss  13876  subrg1  14078  subrgdvds  14082  subrguss  14083  subrginv  14084  islss3  14226  lspsnneg  14267  epttop  14647  metequiv2  15053  limccnpcntop  15232  limccnp2lem  15233  limccnp2cntop  15234  nnsf  16114