Theorem bnj1262 32690

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1262.1	⊢ 𝐴 ⊆ 𝐵
bnj1262.2	⊢ (𝜑 → 𝐶 = 𝐴)

Assertion

Ref	Expression
bnj1262	⊢ (𝜑 → 𝐶 ⊆ 𝐵)

Proof of Theorem bnj1262

Step	Hyp	Ref	Expression
1		bnj1262.2	. 2 ⊢ (𝜑 → 𝐶 = 𝐴)
2		bnj1262.1	. 2 ⊢ 𝐴 ⊆ 𝐵
3	1, 2	eqsstrdi 3971	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ⊆ wss 3883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-in 3890  df-ss 3900

This theorem is referenced by:  bnj229  32764  bnj1128  32870  bnj1145  32873