Theorem bnj1262 32192

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 3969	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1538   ⊆ wss 3881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898

This theorem is referenced by:  bnj229  32266  bnj1128  32372  bnj1145  32375