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Mirrors > Home > MPE Home > Th. List > axc4 | Structured version Visualization version GIF version |
Description: Show that the original
axiom ax-c4 34169 can be derived from ax-4 1737
(alim 1738), ax-10 2019 (hbn1 2020), sp 2053 and propositional calculus. See
ax4fromc4 34179 for the rederivation of ax-4 1737
from ax-c4 34169.
Part of the proof is based on the proof of Lemma 22 of [Monk2] p. 114. (Contributed by NM, 21-May-2008.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
axc4 | ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sp 2053 | . . . 4 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑) | |
2 | 1 | con2i 134 | . . 3 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
3 | hbn1 2020 | . . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑) | |
4 | hbn1 2020 | . . . . 5 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
5 | 4 | con1i 144 | . . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) |
6 | 5 | alimi 1739 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
7 | 2, 3, 6 | 3syl 18 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
8 | alim 1738 | . 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓)) | |
9 | 7, 8 | syl5 34 | 1 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
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 |
This theorem depends on definitions: df-bi 197 df-ex 1705 |
This theorem is referenced by: axc5c4c711 38602 |
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