Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > axc4 | Structured version Visualization version GIF version |
Description: Show that the original
axiom ax-c4 33187 can be derived from ax-4 1728
(alim 1729), ax-10 2006 (hbn1 2007), sp 2041 and propositional calculus. See
ax4fromc4 33197 for the rederivation of ax-4 1728
from ax-c4 33187.
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 2041 | . . . 4 ⊢ (∀𝑥 ¬ ∀𝑥𝜑 → ¬ ∀𝑥𝜑) | |
2 | 1 | con2i 133 | . . 3 ⊢ (∀𝑥𝜑 → ¬ ∀𝑥 ¬ ∀𝑥𝜑) |
3 | hbn1 2007 | . . 3 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑) | |
4 | hbn1 2007 | . . . . 5 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑) | |
5 | 4 | con1i 143 | . . . 4 ⊢ (¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥𝜑) |
6 | 5 | alimi 1730 | . . 3 ⊢ (∀𝑥 ¬ ∀𝑥 ¬ ∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
7 | 2, 3, 6 | 3syl 18 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) |
8 | alim 1729 | . 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥∀𝑥𝜑 → ∀𝑥𝜓)) | |
9 | 7, 8 | syl5 33 | 1 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-12 2034 |
This theorem depends on definitions: df-bi 196 df-ex 1696 |
This theorem is referenced by: axc5c4c711 37624 |
Copyright terms: Public domain | W3C validator |