Theorem 19.3h 1567

Description: A wff may be quantified with a variable not free in it. Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 21-May-2007.)

Hypothesis

Ref	Expression
19.3h.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
19.3h	⊢ (∀𝑥𝜑 ↔ 𝜑)

Proof of Theorem 19.3h

Step	Hyp	Ref	Expression
1		ax-4 1524	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
2		19.3h.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
3	1, 2	impbii 126	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia2 107  ax-ia3 108  ax-4 1524

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  19.27h  1574  19.28h  1576  equsalh  1740  2eu4  2138