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Mirrors > Home > MPE Home > Th. List > Mathboxes > hbexg | Structured version Visualization version GIF version |
Description: Closed form of nfex 2325. Derived from hbexgVD 42140. (Contributed by Alan Sare, 8-Feb-2014.) (Revised by Mario Carneiro, 12-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hbexg | ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa2 2176 | . . 3 ⊢ Ⅎ𝑦∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) | |
2 | sp 2182 | . . . . . . 7 ⊢ (∀𝑦(𝜑 → ∀𝑥𝜑) → (𝜑 → ∀𝑥𝜑)) | |
3 | 2 | alimi 1819 | . . . . . 6 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(𝜑 → ∀𝑥𝜑)) |
4 | nf5 2285 | . . . . . 6 ⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑)) | |
5 | 3, 4 | sylibr 237 | . . . . 5 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥𝜑) |
6 | 1, 5 | nfexd 2330 | . . . 4 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → Ⅎ𝑥∃𝑦𝜑) |
7 | nf5 2285 | . . . 4 ⊢ (Ⅎ𝑥∃𝑦𝜑 ↔ ∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | |
8 | 6, 7 | sylib 221 | . . 3 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
9 | 1, 8 | alrimi 2213 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑦∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
10 | alcom 2162 | . 2 ⊢ (∀𝑦∀𝑥(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑) ↔ ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) | |
11 | 9, 10 | sylib 221 | 1 ⊢ (∀𝑥∀𝑦(𝜑 → ∀𝑥𝜑) → ∀𝑥∀𝑦(∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 Ⅎwnf 1791 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-10 2143 ax-11 2160 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: (None) |
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