Theorem 19.12vv 2381

Description: Special case of 19.12 2362 where its converse holds. See 19.12vvv 2027 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 1972	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
2	1	exbii 1881	. 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
3		nfv 1947	. . . 4 ⊢ Ⅎ𝑥𝜓
4	3	nfal 2358	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
5	4	19.36 2269	. 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
6		19.36v 2026	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
7	6	albii 1852	. . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓))
8		nfv 1947	. . . . 5 ⊢ Ⅎ𝑦𝜑
9	8	nfal 2358	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
10	9	19.21 2246	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
11	7, 10	bitr2i 279	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
12	2, 5, 11	3bitri 300	1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by: (None)