Theorem bnj1131 30112

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 2106	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
4	1, 3	mpbi 219	1 ⊢ 𝜑

Syntax hints:   → wi 4  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by:  bnj1468  30170  bnj1014  30284  bnj1128  30312