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Mirrors > Home > MPE Home > Th. List > axi5r | Structured version Visualization version GIF version |
Description: Converse of ax-c4 (intuitionistic logic axiom ax-i5r). (Contributed by Jim Kingdon, 31-Dec-2017.) |
Ref | Expression |
---|---|
axi5r | ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hba1 2137 | . . 3 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑) | |
2 | hba1 2137 | . . 3 ⊢ (∀𝑥𝜓 → ∀𝑥∀𝑥𝜓) | |
3 | 1, 2 | hbim 2112 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → ∀𝑥𝜓)) |
4 | sp 2041 | . . 3 ⊢ (∀𝑥𝜓 → 𝜓) | |
5 | 4 | imim2i 16 | . 2 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥𝜑 → 𝜓)) |
6 | 3, 5 | alrimih 1741 | 1 ⊢ ((∀𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∀𝑥𝜑 → 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → 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-or 384 df-ex 1696 df-nf 1701 |
This theorem is referenced by: (None) |
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