Theorem a4c 843

Description: Existential introduction with implicit substitution. Compare Lemma 14 of [Tarski] p. 70.

Hypotheses

Ref	Expression
a4c.1	⊢ (φ → ∀xφ)
a4c.2	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
a4c	⊢ (φ → ∃xψ)

Proof of Theorem a4c

Step	Hyp	Ref	Expression
1		a4c.1	. . . . 5 ⊢ (φ → ∀xφ)
2	1	hbne 699	. . . 4 ⊢ (¬ φ → ∀x ¬ φ)
3		a4c.2	. . . . 5 ⊢ (x = y → (φ → ψ))
4	3	con3d 87	. . . 4 ⊢ (x = y → (¬ ψ → ¬ φ))
5	2, 4	a4a 842	. . 3 ⊢ (∀x ¬ ψ → ¬ φ)
6	5	con2i 89	. 2 ⊢ (φ → ¬ ∀x ¬ ψ)
7		df-ex 679	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
8	6, 7	sylibr 175	1 ⊢ (φ → ∃xψ)

Syntax hints:  ¬ wn 1   → wi 2  ∀wal 672  ∃wex 678   = weq 797

This theorem is referenced by:  a4c1 844  a4w 929

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  ax-9 799

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org