Theorem bnj1262 32077

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 4020	1 ⊢ (𝜑 → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   = wceq 1533   ⊆ wss 3935

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-in 3942  df-ss 3951

This theorem is referenced by:  bnj229  32151  bnj1128  32257  bnj1145  32260