Theorem spimeh 1925

Description: Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)

Hypotheses

Ref	Expression
spimeh.1	⊢ (𝜑 → ∀𝑥𝜑)
spimeh.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimeh	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)

Proof of Theorem spimeh

Step	Hyp	Ref	Expression
1		spimeh.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax6ev 1890	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimeh.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1764	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
5	4	19.35i 1806	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
6	1, 5	syl 17	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1481  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by:  bj-spimevw  32657  bj-cbvexiw  32659