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Mirrors > Home > MPE Home > Th. List > nfeud | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the unique existential quantifier. Deduction version of nfeu 2655. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker nfeudw 2652 when possible. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 7-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfeud.1 | ⊢ Ⅎ𝑦𝜑 |
nfeud.2 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfeud | ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfeud.1 | . 2 ⊢ Ⅎ𝑦𝜑 | |
2 | nfeud.2 | . . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
3 | 2 | adantr 484 | . 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
4 | 1, 3 | nfeud2 2651 | 1 ⊢ (𝜑 → Ⅎ𝑥∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1536 Ⅎwnf 1785 ∃!weu 2628 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2142 ax-11 2158 ax-12 2175 ax-13 2379 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-mo 2598 df-eu 2629 |
This theorem is referenced by: nfeu 2655 |
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