Theorem spime 1976

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.)

Hypotheses

Ref	Expression
spime.1	⊢ Ⅎxφ
spime.2	⊢ (x = y → (φ → ψ))

Assertion

Ref	Expression
spime	⊢ (φ → ∃xψ)

Proof of Theorem spime

Step	Hyp	Ref	Expression
1		spime.1	. . . . 5 ⊢ Ⅎxφ
2	1	nfn 1793	. . . 4 ⊢ Ⅎx ¬ φ
3		spime.2	. . . . 5 ⊢ (x = y → (φ → ψ))
4	3	con3d 125	. . . 4 ⊢ (x = y → (¬ ψ → ¬ φ))
5	2, 4	spim 1975	. . 3 ⊢ (∀x ¬ ψ → ¬ φ)
6	5	con2i 112	. 2 ⊢ (φ → ¬ ∀x ¬ ψ)
7		df-ex 1542	. 2 ⊢ (∃xψ ↔ ¬ ∀x ¬ ψ)
8	6, 7	sylibr 203	1 ⊢ (φ → ∃xψ)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1540  ∃wex 1541  Ⅎwnf 1544

This proof depends on 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  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This proof depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is used by:  spimed  1977  spimev  1999