Theorem bnj1131 30858

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

Hypotheses

Ref	Expression
bnj1131.1	⊢ (𝜑 → ∀𝑥𝜑)
bnj1131.2	⊢ ∃𝑥𝜑

Assertion

Ref	Expression
bnj1131	⊢ 𝜑

Proof of Theorem bnj1131

Step	Hyp	Ref	Expression
1		bnj1131.2	. 2 ⊢ ∃𝑥𝜑
2		bnj1131.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
3	2	19.9h 2120	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
4	1, 3	mpbi 220	1 ⊢ 𝜑

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:  bnj1468  30916  bnj1014  31030  bnj1128  31058