Theorem bnj228 30057

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 2913	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylbi 206	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	impcom 445	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∈ wcel 1977  ∀wral 2896

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-ral 2901

This theorem is referenced by:  bnj229  30208  bnj999  30281