Theorem 19.3v 1665

Description: Special case of Theorem 19.3 of [Margaris] p. 89. (Contributed by NM, 1-Aug-2017.) (Revised by Wolf Lammen to remove dependency on ax-8, 4-Dec-2017.)

Assertion

Ref	Expression
19.3v	⊢ (∀xφ ↔ φ)

Distinct variable group:   φ,x

Proof of Theorem 19.3v

Step	Hyp	Ref	Expression
1		alex 1572	. 2 ⊢ (∀xφ ↔ ¬ ∃x ¬ φ)
2		19.9v 1664	. . 3 ⊢ (∃x ¬ φ ↔ ¬ φ)
3	2	con2bii 322	. 2 ⊢ (φ ↔ ¬ ∃x ¬ φ)
4	1, 3	bitr4i 243	1 ⊢ (∀xφ ↔ φ)

Syntax hints:  ¬ wn 3   ↔ wb 176  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  spvw  1666  19.9vOLD  1697