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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-cbvalim | Structured version Visualization version GIF version |
Description: A lemma used to prove bj-cbval 34830 in a weak axiomatization. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
bj-cbvalim | ⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax5e 1915 | . . 3 ⊢ (∃𝑥𝜓 → 𝜓) | |
2 | 1 | ax-gen 1798 | . 2 ⊢ ∀𝑦(∃𝑥𝜓 → 𝜓) |
3 | ax-5 1913 | . 2 ⊢ (∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) | |
4 | bj-cbvalimt 34820 | . . . 4 ⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓))))) | |
5 | 4 | com3l 89 | . . 3 ⊢ (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦∃𝑥𝜒 → (∀𝑦(∃𝑥𝜓 → 𝜓) → (∀𝑥𝜑 → ∀𝑦𝜓))))) |
6 | 5 | com14 96 | . 2 ⊢ (∀𝑦(∃𝑥𝜓 → 𝜓) → ((∀𝑥𝜑 → ∀𝑦∀𝑥𝜑) → (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓))))) |
7 | 2, 3, 6 | mp2 9 | 1 ⊢ (∀𝑦∃𝑥𝜒 → (∀𝑦∀𝑥(𝜒 → (𝜑 → 𝜓)) → (∀𝑥𝜑 → ∀𝑦𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 |
This theorem depends on definitions: df-bi 206 df-ex 1783 |
This theorem is referenced by: bj-cbvalimi 34828 |
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