Theorem bnj228 35191

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 3255	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝜓 → (𝑥 ∈ 𝐴 → 𝜓))
3	1, 2	sylbi 220	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → 𝜓))
4	3	impcom 413	1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝜓)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2146  ∀wral 3081

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-ral 3082

This theorem is used by:  bnj229  35339  bnj999  35413