Theorem bnj1397 35231

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

Hypotheses

Ref	Expression
bnj1397.1	⊢ (𝜑 → ∃𝑥𝜓)
bnj1397.2	⊢ (𝜓 → ∀𝑥𝜓)

Assertion

Ref	Expression
bnj1397	⊢ (𝜑 → 𝜓)

Proof of Theorem bnj1397

Step	Hyp	Ref	Expression
1		bnj1397.1	. 2 ⊢ (𝜑 → ∃𝑥𝜓)
2		bnj1397.2	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
3	2	19.9h 2320	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	1, 3	sylib 221	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1808

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-10 2175  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1809  df-nf 1813

This theorem is used by:  bnj1398  35431  bnj1408  35433  bnj1450  35447  bnj1501  35464