Theorem a4w1 930

Description: Infer existence from a substitution instance.

Hypotheses

Ref	Expression
a4w1.1	⊢ (x = y → (φ ↔ ψ))
a4w1.2	⊢ ψ

Assertion

Ref	Expression
a4w1	⊢ ∃xφ

Distinct variable group(s):   ψ,x

Proof of Theorem a4w1

Step	Hyp	Ref	Expression
1		a4w1.2	. 2 ⊢ ψ
2		a4w1.1	. . . 4 ⊢ (x = y → (φ ↔ ψ))
3	2	biimprd 136	. . 3 ⊢ (x = y → (ψ → φ))
4	3	a4w 929	. 2 ⊢ (ψ → ∃xφ)
5	1, 4	ax-mp 6	1 ⊢ ∃xφ

Syntax hints:   → wi 2   ↔ wb 127  ∃wex 678   = weq 797

This theorem is referenced by:  eirrv 3449

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677  ax-9 799  ax-17 925

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679

metamath.org