Theorem 19.12vv 2316

Description: Special case of 19.12 2303 where its converse holds. See 19.12vvv 2037 for a version with a disjoint variable condition requiring fewer axioms. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

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

Distinct variable groups:   𝜓,𝑥   𝜑,𝑦

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

Proof of Theorem 19.12vv

Step	Hyp	Ref	Expression
1		19.21v 1982	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
2	1	exbii 1892	. 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3		nfv 1957	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2299	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
5	4	19.36 2216	. 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
6		19.36v 2036	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
7	6	albii 1863	. . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓))
8		nfv 1957	. . . . 5 ⊢ Ⅎ𝑦𝜑
9	8	nfal 2299	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
10	9	19.21 2192	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
11	7, 10	bitr2i 268	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
12	2, 5, 11	3bitri 289	1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1599  ∃wex 1823

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-10 2135  ax-11 2150  ax-12 2163

This theorem depends on definitions:  df-bi 199  df-ex 1824  df-nf 1828

This theorem is referenced by: (None)