Theorem 19.12b 33777

Description: Version of 19.12vv 2345 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 2200	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
3	2	exbii 1850	. 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
4		19.12b.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 2317	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.36 2223	. 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
7	4	19.36 2223	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
8	7	albii 1822	. . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓))
9	1	nfal 2317	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
10	9	19.21 2200	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
11	8, 10	bitr2i 275	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
12	3, 6, 11	3bitri 297	1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782  Ⅎwnf 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-ex 1783  df-nf 1787

This theorem is referenced by: (None)