Theorem syl5eqssr 2990

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 2044	. 2 ⊢ 𝐴 = 𝐵
3		syl5eqssr.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
4	2, 3	syl5eqss 2989	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1243   ⊆ wss 2917

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-11 1397  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-in 2924  df-ss 2931

This theorem is referenced by:  relcnvtr  4840  resasplitss  5069  fimacnvdisj  5074  fimacnv  5296  f1ompt  5320