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Mirrors > Home > MPE Home > Th. List > nfeu1ALT | Structured version Visualization version GIF version |
Description: Alternate proof of nfeu1 2674. This illustrates the systematic way of proving nonfreeness in a defined expression: consider the definiens as a tree whose nodes are its subformulas, and prove by tree-induction nonfreeness of each node, starting from the leaves (generally using nfv 1915 or nf* theorems for previously defined expressions) and up to the root. Here, the definiens is a conjunction of two previously defined expressions, which automatically yields the present proof. (Contributed by BJ, 2-Oct-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfeu1ALT | ⊢ Ⅎ𝑥∃!𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-eu 2654 | . 2 ⊢ (∃!𝑥𝜑 ↔ (∃𝑥𝜑 ∧ ∃*𝑥𝜑)) | |
2 | nfe1 2154 | . . 3 ⊢ Ⅎ𝑥∃𝑥𝜑 | |
3 | nfmo1 2641 | . . 3 ⊢ Ⅎ𝑥∃*𝑥𝜑 | |
4 | 2, 3 | nfan 1900 | . 2 ⊢ Ⅎ𝑥(∃𝑥𝜑 ∧ ∃*𝑥𝜑) |
5 | 1, 4 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥∃!𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∃wex 1780 Ⅎwnf 1784 ∃*wmo 2620 ∃!weu 2653 |
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-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-mo 2622 df-eu 2654 |
This theorem is referenced by: (None) |
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