Theorem bnj1241 32074

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

Hypotheses

Ref	Expression
bnj1241.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
bnj1241.2	⊢ (𝜓 → 𝐶 = 𝐴)

Assertion

Ref	Expression
bnj1241	⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵)

Proof of Theorem bnj1241

Step	Hyp	Ref	Expression
1		bnj1241.2	. . . 4 ⊢ (𝜓 → 𝐶 = 𝐴)
2	1	eqcomd 2827	. . 3 ⊢ (𝜓 → 𝐴 = 𝐶)
3	2	adantl 484	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
4		bnj1241.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
5	4	adantr 483	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐵)
6	3, 5	eqsstrrd 4005	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 398   = 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:  bnj1245  32281  bnj1311  32291