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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 2375. See nfrexw 3311 for a version with a disjoint variable condition, but not requiring ax-13 2375. (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 1801 | . . 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 3371 | . 2 ⊢ (⊤ → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑) |
7 | 6 | mptru 1544 | 1 ⊢ Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ⊤wtru 1538 Ⅎwnf 1780 Ⅎwnfc 2888 ∃wrex 3068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-13 2375 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1540 df-ex 1777 df-nf 1781 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 |
This theorem is referenced by: nfiung 5030 |
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