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Description: Alternate proof of nfeu1 . 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 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 | ||
|---|---|---|---|
| Assertion | nfeu1ALT | |- F/ x E! x ph |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-eu | |- ( E! x ph <-> ( E. x ph /\ E* x ph ) ) |
|
| 2 | nfe1 | |- F/ x E. x ph |
|
| 3 | nfmo1 | |- F/ x E* x ph |
|
| 4 | 2 3 | nfan | |- F/ x ( E. x ph /\ E* x ph ) |
| 5 | 1 4 | nfxfr | |- F/ x E! x ph |