Theorem syl6eqssr 3077

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 2093	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
3		syl6eqssr.2	. 2 ⊢ 𝐵 ⊆ 𝐶
4	2, 3	syl6eqss 3076	1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1289   ⊆ wss 2999

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 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070

This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3005  df-ss 3012

This theorem is referenced by:  ffvresb  5455  tposss  6003  sbthlemi5  6660  iooval2  9323  telfsumo  10847