Theorem 19.12b 36291

Description: Version of 19.12vv 2379 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 2243	. . 3 ⊢ (∀𝑦(𝜑 → 𝜓) ↔ (𝜑 → ∀𝑦𝜓))
3	2	exbii 1878	. 2 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∃𝑥(𝜑 → ∀𝑦𝜓))
4		19.12b.2	. . . 4 ⊢ Ⅎ𝑥𝜓
5	4	nfal 2356	. . 3 ⊢ Ⅎ𝑥∀𝑦𝜓
6	5	19.36 2266	. 2 ⊢ (∃𝑥(𝜑 → ∀𝑦𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
7	4	19.36 2266	. . . 4 ⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → 𝜓))
8	7	albii 1849	. . 3 ⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) ↔ ∀𝑦(∀𝑥𝜑 → 𝜓))
9	1	nfal 2356	. . . 4 ⊢ Ⅎ𝑦∀𝑥𝜑
10	9	19.21 2243	. . 3 ⊢ (∀𝑦(∀𝑥𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∀𝑦𝜓))
11	8, 10	bitr2i 279	. 2 ⊢ ((∀𝑥𝜑 → ∀𝑦𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))
12	3, 6, 11	3bitri 300	1 ⊢ (∃𝑥∀𝑦(𝜑 → 𝜓) ↔ ∀𝑦∃𝑥(𝜑 → 𝜓))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1568  ∃wex 1809  Ⅎwnf 1813

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-10 2176  ax-11 2192  ax-12 2213

This theorem depends on definitions:  df-bi 210  df-ex 1810  df-nf 1814

This theorem is referenced by: (None)