Theorem spimeh 1667

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 = z → (φ → ψ))

Assertion

Ref	Expression
spimeh	⊢ (φ → ∃xψ)

Distinct variable group:   x,z

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

Proof of Theorem spimeh

Step	Hyp	Ref	Expression
1		spimeh.1	. 2 ⊢ (φ → ∀xφ)
2		a9ev 1656	. . . 4 ⊢ ∃x x = z
3		spimeh.2	. . . . 5 ⊢ (x = z → (φ → ψ))
4	3	eximi 1576	. . . 4 ⊢ (∃x x = z → ∃x(φ → ψ))
5	2, 4	ax-mp 5	. . 3 ⊢ ∃x(φ → ψ)
6	5	19.35i 1601	. 2 ⊢ (∀xφ → ∃xψ)
7	1, 6	syl 15	1 ⊢ (φ → ∃xψ)

Syntax hints:   → wi 4  ∀wal 1540  ∃wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-9 1654

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by:  ax12olem1  1927  ax10lem2  1937