Theorem 19.12 729

Description: Theorem 19.12 of [Margaris] p. 89. Assuming the converse is a mistake sometimes made by beginners! But sometimes the converse does hold, as in 19.12vv 960.

Assertion

Ref	Expression
19.12	⊢ (∃x∀yφ → ∀y∃xφ)

Proof of Theorem 19.12

Step	Hyp	Ref	Expression
1		hba1 698	. . 3 ⊢ (∀yφ → ∀y∀yφ)
2	1	hbex 701	. 2 ⊢ (∃x∀yφ → ∀y∃x∀yφ)
3		ax-4 673	. . . 4 ⊢ (∀yφ → φ)
4	3	19.22i 723	. . 3 ⊢ (∃x∀yφ → ∃xφ)
5	4	19.20i 691	. 2 ⊢ (∀y∃x∀yφ → ∀y∃xφ)
6	2, 5	syl 12	1 ⊢ (∃x∀yφ → ∀y∃xφ)

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

This theorem is referenced by:  hbexd 791  iinss 2025

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-7 676  ax-gen 677

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

metamath.org