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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax10fromc7 | Structured version Visualization version GIF version |
Description: Rederivation of axiom ax-10 2174 from ax-c7 34686, ax-c4 34685, ax-c5 34684, ax-gen 1870 and propositional calculus. See axc7 2296 for the derivation of ax-c7 34686 from ax-10 2174. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ax10fromc7 | ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c4 34685 | . . 3 ⊢ (∀𝑥(∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)) | |
2 | ax-c5 34684 | . . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥∀𝑥𝜑) | |
3 | ax-c4 34685 | . . . . 5 ⊢ (∀𝑥(∀𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)) | |
4 | id 22 | . . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥𝜑) | |
5 | 3, 4 | mpg 1872 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
6 | 2, 5 | nsyl 137 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) |
7 | 1, 6 | mpg 1872 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
8 | ax-c7 34686 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
9 | 7, 8 | nsyl4 157 | 1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-c5 34684 ax-c4 34685 ax-c7 34686 |
This theorem is referenced by: hba1-o 34698 axc5c711 34719 equidq 34725 |
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