Theorem syl5eqssr 3053

Description: B chained subclass and equality deduction. (Contributed by NM, 25-Apr-2004.)

Hypotheses

Ref	Expression
syl5eqssr.1	⊢ 𝐵 = 𝐴
syl5eqssr.2	⊢ (𝜑 → 𝐵 ⊆ 𝐶)

Assertion

Ref	Expression
syl5eqssr	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl5eqssr

Step	Hyp	Ref	Expression
1		syl5eqssr.1	. . 3 ⊢ 𝐵 = 𝐴
2	1	eqcomi 2087	. 2 ⊢ 𝐴 = 𝐵
3		syl5eqssr.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	syl5eqss 3052	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1285   ⊆ wss 2982

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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-in 2988  df-ss 2995

This theorem is referenced by:  relcnvtr  4890  resasplitss  5120  fimacnvdisj  5125  fimacnv  5349  f1ompt  5373  tfr1onlemres  6019  tfrcllemres  6032