Theorem bnj1241 34944

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 2743	. . 3 ⊢ (𝜓 → 𝐴 = 𝐶)
3	2	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 = 𝐶)
4		bnj1241.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
5	4	adantr 480	. 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐴 ⊆ 𝐵)
6	3, 5	eqsstrrd 3970	1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 ⊆ 𝐵)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ⊆ wss 3902

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 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-cleq 2729  df-ss 3919

This theorem is referenced by:  bnj1245  35151  bnj1311  35161