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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-wnfnf | Structured version Visualization version GIF version |
Description: When 𝜑 is substituted for 𝜓, this statement expresses nonfreeness in the weak form of nonfreeness (∃ → ∀). Note that this could also be proved from bj-nnfim 34928, bj-nnfe1 34942 and bj-nnfa1 34941. (Contributed by BJ, 9-Dec-2023.) |
Ref | Expression |
---|---|
bj-wnfnf | ⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-wnf2 34900 | . 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) | |
2 | bj-wnf1 34899 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) | |
3 | df-bj-nnf 34906 | . 2 ⊢ (Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) ↔ ((∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) ∧ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)))) | |
4 | 1, 2, 3 | mpbir2an 708 | 1 ⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1537 ∃wex 1782 Ⅎ'wnnf 34905 |
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 ax-6 1971 ax-7 2011 ax-10 2137 ax-12 2171 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-ex 1783 df-nf 1787 df-bj-nnf 34906 |
This theorem is referenced by: (None) |
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