Theorem syl6ssr 1547

Description: A chained subclass and equality deduction.

Hypotheses

Ref	Expression
syl6ssr.1	⊢ (φ → A ⊆ B)
syl6ssr.2	⊢ C = B

Assertion

Ref	Expression
syl6ssr	⊢ (φ → A ⊆ C)

Proof of Theorem syl6ssr

Step	Hyp	Ref	Expression
1		syl6ssr.1	. 2 ⊢ (φ → A ⊆ B)
2		syl6ssr.2	. . 3 ⊢ C = B
3	2	cleqcomi 1105	. 2 ⊢ B = C
4	1, 3	syl6ss 1546	1 ⊢ (φ → A ⊆ C)

Syntax hints:   → wi 2   = wceq 1091   ⊆ wss 1487

This theorem is referenced by:  iunpw 2040  tfrlem9 2957  tfrlem13 2961  tz7.49 2997  cplem1 3545  zornlem2 3604  infxpidmlem5 4937  mdsymlem1 5776

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492

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