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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 34691, bj-nnfe1 34705 and bj-nnfa1 34704. (Contributed by BJ, 9-Dec-2023.) |
Ref | Expression |
---|---|
bj-wnfnf | ⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-wnf2 34663 | . 2 ⊢ (∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) | |
2 | bj-wnf1 34662 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) | |
3 | df-bj-nnf 34669 | . 2 ⊢ (Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) ↔ ((∃𝑥(∃𝑥𝜑 → ∀𝑥𝜓) → (∃𝑥𝜑 → ∀𝑥𝜓)) ∧ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)))) | |
4 | 1, 2, 3 | mpbir2an 711 | 1 ⊢ Ⅎ'𝑥(∃𝑥𝜑 → ∀𝑥𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 Ⅎ'wnnf 34668 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2142 ax-12 2176 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-ex 1788 df-nf 1792 df-bj-nnf 34669 |
This theorem is referenced by: (None) |
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