Theorem bnj1397 30905

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 2120	. 2 ⊢ (∃𝑥𝜓 ↔ 𝜓)
4	1, 3	sylib 208	1 ⊢ (𝜑 → 𝜓)

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nf 1710

This theorem is referenced by:  bnj1398  31102  bnj1408  31104  bnj1450  31118  bnj1501  31135