Theorem bnj1459 32723

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

Hypotheses

Ref	Expression
bnj1459.1	⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴))
bnj1459.2	⊢ (𝜓 → 𝜒)

Assertion

Ref	Expression
bnj1459	⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜒)

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem bnj1459

Step	Hyp	Ref	Expression
1		bnj1459.1	. . 3 ⊢ (𝜓 ↔ (𝜑 ∧ 𝑥 ∈ 𝐴))
2		bnj1459.2	. . 3 ⊢ (𝜓 → 𝜒)
3	1, 2	sylbir 234	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜒)
4	3	ralrimiva 3107	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜒)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∈ wcel 2108  ∀wral 3063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914

This theorem depends on definitions:  df-bi 206  df-an 396  df-ral 3068

This theorem is referenced by:  bnj1501  32947