Theorem 19.9r 718

Description: Variation of Theorem 19.9 of [Margaris] p. 89.

Hypothesis

Ref	Expression
19.9r.1	⊢ (φ → ∀xφ)

Assertion

Ref	Expression
19.9r	⊢ (φ ↔ ∃xφ)

Proof of Theorem 19.9r

Step	Hyp	Ref	Expression
1		19.8a 712	. 2 ⊢ (φ → ∃xφ)
2		df-ex 679	. . 3 ⊢ (∃xφ ↔ ¬ ∀x ¬ φ)
3		19.9r.1	. . . . . . 7 ⊢ (φ → ∀xφ)
4	3	con3i 90	. . . . . 6 ⊢ (¬ ∀xφ → ¬ φ)
5	4	19.20i 691	. . . . 5 ⊢ (∀x ¬ ∀xφ → ∀x ¬ φ)
6	5	con3i 90	. . . 4 ⊢ (¬ ∀x ¬ φ → ¬ ∀x ¬ ∀xφ)
7		ax-6 675	. . . 4 ⊢ (¬ ∀x ¬ ∀xφ → φ)
8	6, 7	syl 12	. . 3 ⊢ (¬ ∀x ¬ φ → φ)
9	2, 8	sylbi 174	. 2 ⊢ (∃xφ → φ)
10	1, 9	impbi 139	1 ⊢ (φ ↔ ∃xφ)

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678

This theorem is referenced by:  excomim 727  19.19 737  19.23 745  19.23ai 746  19.36 757  19.44 768  19.45 769  19.9rv 941  exists1 1072

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-ex 679

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