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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax10fromc7 | Structured version Visualization version GIF version |
Description: Rederivation of Axiom ax-10 2137 from ax-c7 37750, ax-c4 37749, ax-c5 37748, ax-gen 1797 and propositional calculus. See axc7 2310 for the derivation of ax-c7 37750 from ax-10 2137. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-10 2137 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
ax10fromc7 | ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c4 37749 | . . 3 ⊢ (∀𝑥(∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)) | |
2 | ax-c5 37748 | . . . 4 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥∀𝑥𝜑) | |
3 | ax-c4 37749 | . . . . 5 ⊢ (∀𝑥(∀𝑥𝜑 → ∀𝑥𝜑) → (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)) | |
4 | id 22 | . . . . 5 ⊢ (∀𝑥𝜑 → ∀𝑥𝜑) | |
5 | 3, 4 | mpg 1799 | . . . 4 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
6 | 2, 5 | nsyl 140 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ¬ ∀𝑥𝜑) |
7 | 1, 6 | mpg 1799 | . 2 ⊢ (∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
8 | ax-c7 37750 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥∀𝑥𝜑 → ∀𝑥𝜑) | |
9 | 7, 8 | nsyl4 158 | 1 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1539 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-c5 37748 ax-c4 37749 ax-c7 37750 |
This theorem is referenced by: hba1-o 37762 axc5c711 37783 equidq 37789 |
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