Theorem 19.8wOLD 1693

Description: Obsolete version of 19.8w 1660 as of 4-Dec-2017. (Contributed by NM, 1-Aug-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
19.8wOLD.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.8wOLD	⊢ (φ → ∃xφ)

Proof of Theorem 19.8wOLD

Step	Hyp	Ref	Expression
1		19.8wOLD.1	. . . . 5 ⊢ (φ → ∀xφ)
2		notnot 282	. . . . 5 ⊢ (φ ↔ ¬ ¬ φ)
3	2	albii 1566	. . . . 5 ⊢ (∀xφ ↔ ∀x ¬ ¬ φ)
4	1, 2, 3	3imtr3i 256	. . . 4 ⊢ (¬ ¬ φ → ∀x ¬ ¬ φ)
5	4	spnfw 1670	. . 3 ⊢ (∀x ¬ φ → ¬ φ)
6	5	con2i 112	. 2 ⊢ (φ → ¬ ∀x ¬ φ)
7		df-ex 1542	. 2 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
8	6, 7	sylibr 203	1 ⊢ (φ → ∃xφ)

Syntax hints:  ¬ wn 3   → wi 4  ∀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-9 1654

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

This theorem is referenced by: (None)