Theorem bnj1294

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

	Ref	Expression
Hypotheses	bnj1294.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
	bnj1294.2	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
Assertion	bnj1294	⊢ ( 𝜑 → 𝜓 )

Proof

Step	Hyp	Ref	Expression
1		bnj1294.1	⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝜓 )
2		bnj1294.2	⊢ ( 𝜑 → 𝑥 ∈ 𝐴 )
3		df-ral	⊢ ( ∀ 𝑥 ∈ 𝐴 𝜓 ↔ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) )
4		sp	⊢ ( ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) → ( 𝑥 ∈ 𝐴 → 𝜓 ) )
5	4	impcom	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑥 ( 𝑥 ∈ 𝐴 → 𝜓 ) ) → 𝜓 )
6	3 5	sylan2b	⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∀ 𝑥 ∈ 𝐴 𝜓 ) → 𝜓 )
7	2 1 6	syl2anc	⊢ ( 𝜑 → 𝜓 )