Theorem speiv 2000

Description: Inference from existential specialization, using implicit substitution. (Contributed by NM, 19-Aug-1993.)

Hypotheses

Ref	Expression
speiv.1	⊢ (x = y → (φ ↔ ψ))
speiv.2	⊢ ψ

Assertion

Ref	Expression
speiv	⊢ ∃xφ

Distinct variable group:   ψ,x

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

Proof of Theorem speiv

Step	Hyp	Ref	Expression
1		speiv.2	. 2 ⊢ ψ
2		speiv.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	biimprd 214	. . 3 ⊢ (x = y → (ψ → φ))
4	3	spimev 1999	. 2 ⊢ (ψ → ∃xφ)
5	1, 4	ax-mp 5	1 ⊢ ∃xφ

Syntax hints:   → wi 4   ↔ wb 176  ∃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-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  ce0addcnnul  6180