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Mirrors > Home > MPE Home > Th. List > Mathboxes > ax4fromc4 | Structured version Visualization version GIF version |
Description: Rederivation of Axiom ax-4 1817 from ax-c4 36635, ax-c5 36634, ax-gen 1803 and minimal implicational calculus { ax-mp 5, ax-1 6, ax-2 7 }. See axc4 2320 for the derivation of ax-c4 36635 from ax-4 1817. (Contributed by NM, 23-May-2008.) (Proof modification is discouraged.) Use ax-4 1817 instead. (New usage is discouraged.) |
Ref | Expression |
---|---|
ax4fromc4 | ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-c4 36635 | . . 3 ⊢ (∀𝑥(∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) → (∀𝑥(𝜑 → 𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓))) | |
2 | ax-c5 36634 | . . . 4 ⊢ (∀𝑥𝜑 → 𝜑) | |
3 | ax-c5 36634 | . . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓)) | |
4 | 2, 3 | syl5 34 | . . 3 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓)) |
5 | 1, 4 | mpg 1805 | . 2 ⊢ (∀𝑥(𝜑 → 𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) |
6 | ax-c4 36635 | . 2 ⊢ (∀𝑥(∀𝑥𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | |
7 | 5, 6 | syl 17 | 1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-gen 1803 ax-c5 36634 ax-c4 36635 |
This theorem is referenced by: (None) |
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