Theorem syl6eqssr 2996

Description: A chained subclass and equality deduction. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
syl6eqssr.1	⊢ (𝜑 → 𝐵 = 𝐴)
syl6eqssr.2	⊢ 𝐵 ⊆ 𝐶

Assertion

Ref	Expression
syl6eqssr	⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Proof of Theorem syl6eqssr

Step	Hyp	Ref	Expression
1		syl6eqssr.1	. . 3 ⊢ (𝜑 → 𝐵 = 𝐴)
2	1	eqcomd 2045	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		syl6eqssr.2	. 2 ⊢ 𝐵 ⊆ 𝐶
4	2, 3	syl6eqss 2995	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:  ffvresb  5328  tposss  5861  iooval2  8784