Theorem 19.9h 2317

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 2197	. 2 ⊢ Ⅎ𝑥𝜑
3	2	19.9 2247	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654  ∃wex 1878

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220

This theorem depends on definitions:  df-bi 199  df-ex 1879  df-nf 1883

This theorem is referenced by:  bnj1131  31393  bnj1397  31440