Mathbox for Wolf Lammen |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > wl-nfimf1 | Structured version Visualization version GIF version |
Description: An antecedent is irrelevant to a not-free property, if it always holds. I used this variant of nfim 1897 in dvelimdf 2471 to simplify the proof. (Contributed by Wolf Lammen, 14-Oct-2018.) |
Ref | Expression |
---|---|
wl-nfimf1 | ⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfa1 2155 | . 2 ⊢ Ⅎ𝑥∀𝑥𝜑 | |
2 | pm5.5 364 | . . 3 ⊢ (𝜑 → ((𝜑 → 𝜓) ↔ 𝜓)) | |
3 | 2 | sps 2184 | . 2 ⊢ (∀𝑥𝜑 → ((𝜑 → 𝜓) ↔ 𝜓)) |
4 | 1, 3 | nfbidf 2226 | 1 ⊢ (∀𝑥𝜑 → (Ⅎ𝑥(𝜑 → 𝜓) ↔ Ⅎ𝑥𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 Ⅎwnf 1784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-or 844 df-ex 1781 df-nf 1785 |
This theorem is referenced by: (None) |
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