Theorem bnj228 31346

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj228.1	⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)

Assertion

Ref	Expression
bnj228	⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)

Proof of Theorem bnj228

Step	Hyp	Ref	Expression
1		bnj228.1	. . 3 ⊢ (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜓)
2		rsp 3138	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylbi 209	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	impcom 398	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2164  ∀wral 3117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-12 2220

This theorem depends on definitions:  df-bi 199  df-an 387  df-ex 1879  df-ral 3122

This theorem is referenced by:  bnj229  31496  bnj999  31569