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Mirrors > Home > MPE Home > Th. List > axc10 | Structured version Visualization version GIF version |
Description: Show that the original
axiom ax-c10 34171 can be derived from ax6 2251
and axc7 2132
(on top of propositional calculus, ax-gen 1722, and ax-4 1737). See
ax6fromc10 34181 for the rederivation of ax6 2251
from ax-c10 34171.
Normally, axc10 2252 should be used rather than ax-c10 34171, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
axc10 | ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6 2251 | . . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | |
2 | con3 149 | . . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦)) | |
3 | 2 | al2imi 1743 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦)) |
4 | 1, 3 | mtoi 190 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
5 | axc7 2132 | . 2 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → 𝜑) | |
6 | 4, 5 | syl 17 | 1 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1481 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 ax-13 2246 |
This theorem depends on definitions: df-bi 197 df-an 386 df-ex 1705 |
This theorem is referenced by: (None) |
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