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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-wnfanf | Structured version Visualization version GIF version |
Description: When 𝜑 is substituted for 𝜓, this statement expresses that weak nonfreeness implies the universal form of nonfreeness. (Contributed by BJ, 9-Dec-2023.) |
Ref | Expression |
---|---|
bj-wnfanf | ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-wnf1 34667 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(∃𝑥𝜑 → ∀𝑥𝜓)) | |
2 | bj-19.23bit 34641 | . 2 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜓)) | |
3 | 1, 2 | sylg 1830 | 1 ⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → ∀𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1541 ∃wex 1787 |
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 2143 ax-12 2177 |
This theorem depends on definitions: df-bi 210 df-or 848 df-ex 1788 df-nf 1792 |
This theorem is referenced by: (None) |
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