Theorem spime 2421

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. See spimew 2001 and spimevw 2015 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 2404. Use spimefv 2234 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 2420	. 2 ⊢ (⊤ → (𝜑 → ∃𝑥𝜓))
5	4	mptru 1577	1 ⊢ (𝜑 → ∃𝑥𝜓)

This proof depends on syntax axioms:   → wi 4  ⊤wtru 1571  ∃wex 1809  Ⅎwnf 1813

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-12 2213  ax-13 2404

This proof depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-nf 1814

This theorem is used by:  spimev  2424  exnel  36300