Theorem 19.12vv 2168

Description: Special case of 19.12 2150 where its converse holds. See 19.12vvv 1894 for a version with a dv 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 1855	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
2	1	exbii 1764	. 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3		nfv 1830	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2139	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
5	4	19.36 2085	. 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
6		19.36v 1891	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
7	6	albii 1737	. . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓))
8		nfv 1830	. . . . 5 ⊢ Ⅎ𝑦𝜑
9	8	nfal 2139	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
10	9	19.21 2062	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
11	7, 10	bitr2i 264	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
12	2, 5, 11	3bitri 285	1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)