Theorem bnj1262 34786

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 4063	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1537   ⊆ wss 3976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1778  df-cleq 2732  df-ss 3993

This theorem is referenced by:  bnj229  34860  bnj1128  34966  bnj1145  34969