Theorem spime 2423

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 2004 and spimevw 2018 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 2406. Use spimefv 2237 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 2422	. 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓))
5	4	mptru 1577	1 ⊢ (𝜑 → ∃𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ⊤wtru 1571  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-nf 1817

This theorem is used by:  spimev  2426  exnel  36334