Theorem 19.9h 2297

Description: A wff may be existentially quantified with a variable not free in it. Theorem 19.9 of [Margaris] p. 89. (Contributed by FL, 24-Mar-2007.) (Proof shortened by Wolf Lammen, 5-Jan-2018.) (Proof shortened by Wolf Lammen, 14-Jul-2020.)

Hypothesis

Ref	Expression
19.9h.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
19.9h	⊢ (∃𝑥𝜑 ↔ 𝜑)

Proof of Theorem 19.9h

Step	Hyp	Ref	Expression
1		19.9h.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2	1	nf5i 2157	. 2 ⊢ Ⅎ𝑥𝜑
3	2	19.9 2217	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by:  bnj1131  34970  bnj1397  35016