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| Mirrors > Home > MPE Home > Th. List > nfrex | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2377. See nfrexw 3313 for a version with a disjoint variable condition, but not requiring ax-13 2377. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) (New usage is discouraged.) |
| Ref | Expression |
|---|---|
| nfral.1 | ⊢ Ⅎ𝑥𝐴 |
| nfral.2 | ⊢ Ⅎ𝑥𝜑 |
| Ref | Expression |
|---|---|
| nfrex | ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nftru 1804 | . . 3 ⊢ Ⅎ𝑦⊤ | |
| 2 | nfral.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 3 | 2 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝐴) |
| 4 | nfral.2 | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
| 5 | 4 | a1i 11 | . . 3 ⊢ (⊤ → Ⅎ𝑥𝜑) |
| 6 | 1, 3, 5 | nfrexd 3373 | . 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑) |
| 7 | 6 | mptru 1547 | 1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
| Colors of variables: wff setvar class |
| Syntax hints: ⊤wtru 1541 Ⅎwnf 1783 Ⅎwnfc 2890 ∃wrex 3070 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-13 2377 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 |
| This theorem is referenced by: nfiung 5025 |
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