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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 36182 can be derived from ax6 2391
and axc7 2325
(on top of propositional calculus, ax-gen 1797, and ax-4 1811). See
ax6fromc10 36192 for the rederivation of ax6 2391
from ax-c10 36182.
Normally, axc10 2392 should be used rather than ax-c10 36182, except by theorems specifically studying the latter's properties. Usage of this theorem has been discouraged later on to avoid ax-13 2379 propagation. Check out bj-axc10v 34230 for a weaker version requiring less axioms. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
axc10 | ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax6 2391 | . . 3 ⊢ ¬ ∀𝑥 ¬ 𝑥 = 𝑦 | |
2 | con3 156 | . . . 4 ⊢ ((𝑥 = 𝑦 → ∀𝑥𝜑) → (¬ ∀𝑥𝜑 → ¬ 𝑥 = 𝑦)) | |
3 | 2 | al2imi 1817 | . . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → (∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝑥 = 𝑦)) |
4 | 1, 3 | mtoi 202 | . 2 ⊢ (∀𝑥(𝑥 = 𝑦 → ∀𝑥𝜑) → ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
5 | axc7 2325 | . 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 1536 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: spALT 40907 |
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