Theorem bnj1262 34803

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 4050	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ⊆ wss 3963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1777  df-cleq 2727  df-ss 3980

This theorem is referenced by:  bnj229  34877  bnj1128  34983  bnj1145  34986