Theorem bnj1131 35418

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 2319	. 2 ⊢ (∃𝑥𝜑 ↔ 𝜑)
4	1, 3	mpbi 233	1 ⊢ 𝜑

This proof depends on syntax axioms:   → wi 4  ∀wal 1567  ∃wex 1811

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-10 2177  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by:  bnj1468  35476  bnj1014  35591  bnj1128  35620