Theorem bnj1397 35531

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 2323	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	1, 3	sylib 221	1 ⊢ (𝜑 → 𝜓)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  ∃wex 1812

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 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by:  bnj1398  35731  bnj1408  35733  bnj1450  35747  bnj1501  35764