Theorem bnj1459 35473

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 238	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜒)
4	3	ralrimiva 3154	1 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜒)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2144  ∀wral 3076

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942

This proof depends on definitions:  df-bi 210  df-an 402  df-ral 3077

This theorem is used by:  bnj1501  35697