Theorem spime 2389

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 1972 and spimevw 1986 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 2372. Use spimefv 2201 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 2388	. 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓))
5	4	mptru 1548	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ⊤wtru 1542  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-12 2180  ax-13 2372

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-nf 1785

This theorem is referenced by:  spimev  2392  exnel  35844