Theorem bnj228 32714

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 3131	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylbi 216	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	impcom 408	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∈ wcel 2106  ∀wral 3064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-ex 1783  df-ral 3069

This theorem is referenced by:  bnj229  32864  bnj999  32938