Theorem 19.12b 33048

Description: Version of 19.12vv 2368 with not-free hypotheses, instead of distinct variable conditions. (Contributed by Scott Fenton, 13-Dec-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
19.12b.1	⊢ Ⅎ𝑦𝜑
19.12b.2	⊢ Ⅎ𝑥𝜓

Assertion

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

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem 19.12b

Step	Hyp	Ref	Expression
1		19.12b.1	. . . 4 ⊢ Ⅎ𝑦𝜑
2	1	19.21 2207	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
3	2	exbii 1848	. 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
4		19.12b.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 2342	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.36 2232	. 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
7	4	19.36 2232	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
8	7	albii 1820	. . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓))
9	1	nfal 2342	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
10	9	19.21 2207	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
11	8, 10	bitr2i 278	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
12	3, 6, 11	3bitri 299	1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1535  ∃wex 1780  Ⅎwnf 1784

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)