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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-wnf2 | Structured version Visualization version GIF version |
Description: When 𝜑 is substituted for 𝜓, this is the first half of nonfreness (. → ∀) of the weak form of nonfreeness (∃ → ∀). (Contributed by BJ, 9-Dec-2023.) |
Ref | Expression |
---|---|
bj-wnf2 | ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbe1 2143 | . 2 ⊢ (∃𝑥𝜑 → ∀𝑥∃𝑥𝜑) | |
2 | bj-eximcom 34818 | . 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∀𝑥∃𝑥𝜑 → ∃𝑥∀𝑥𝜓)) | |
3 | hbe1a 2144 | . 2 ⊢ (∃𝑥∀𝑥𝜓 → ∀𝑥𝜓) | |
4 | 1, 2, 3 | syl56 36 | 1 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 ∃wex 1786 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-10 2141 |
This theorem depends on definitions: df-bi 206 df-ex 1787 |
This theorem is referenced by: bj-wnfnf 34915 |
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