Theorem spime 2384

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1968 and spimevw 1991 for weaker versions requiring fewer axioms. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.) Usage of this theorem is discouraged because it depends on ax-13 2367. Use spimefv 2187 instead. (New usage is discouraged.)

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	. . . 4 ⊢ Ⅎ𝑥𝜑
2	1	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
3		spime.2	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	spimed 2383	. 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓))
5	4	mptru 1541	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ⊤wtru 1535  ∃wex 1774  Ⅎwnf 1778

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-12 2167  ax-13 2367

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-ex 1775  df-nf 1779

This theorem is referenced by:  spimev  2387  exnel  35392