Theorem bnj1131 32642

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 2289	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
4	1, 3	mpbi 233	1 ⊢ 𝜑

Syntax hints:   → wi 4  ∀wal 1541  ∃wex 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-10 2143  ax-12 2177

This theorem depends on definitions:  df-bi 210  df-ex 1788  df-nf 1792

This theorem is referenced by:  bnj1468  32701  bnj1014  32816  bnj1128  32845