Theorem 19.12vv 960

Description: Special case of 19.12 729 where its converse holds.

Assertion

Ref	Expression
19.12vv	⊢ (∃x∀y(φ → ψ) ↔ ∀y∃x(φ → ψ))

Distinct variable group(s):   x,y   ψ,x   φ,y

Proof of Theorem 19.12vv

Step	Hyp	Ref	Expression
1		19.21v 942	. . 3 ⊢ (∀y(φ → ψ) ↔ (φ → ∀yψ))
2	1	biex 733	. 2 ⊢ (∃x∀y(φ → ψ) ↔ ∃x(φ → ∀yψ))
3		19.36v 958	. 2 ⊢ (∃x(φ → ∀yψ) ↔ (∀xφ → ∀yψ))
4		19.36v 958	. . . 4 ⊢ (∃x(φ → ψ) ↔ (∀xφ → ψ))
5	4	bial 695	. . 3 ⊢ (∀y∃x(φ → ψ) ↔ ∀y(∀xφ → ψ))
6		19.21v 942	. . 3 ⊢ (∀y(∀xφ → ψ) ↔ (∀xφ → ∀yψ))
7	5, 6	bitr2 152	. 2 ⊢ ((∀xφ → ∀yψ) ↔ ∀y∃x(φ → ψ))
8	2, 3, 7	3bitr 155	1 ⊢ (∃x∀y(φ → ψ) ↔ ∀y∃x(φ → ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678

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-17 925

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

metamath.org