![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > nfrexd | Structured version Visualization version GIF version |
Description: Deduction version of nfrex 3347. Usage of this theorem is discouraged because it depends on ax-13 2371. See nfrexdw 3292 for a version with a disjoint variable condition, but not requiring ax-13 2371. (Contributed by Mario Carneiro, 14-Oct-2016.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfrald.1 | ⊢ Ⅎ𝑦𝜑 |
nfrald.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfrald.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrexd | ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfrex2 3073 | . 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) | |
2 | nfrald.1 | . . . 4 ⊢ Ⅎ𝑦𝜑 | |
3 | nfrald.2 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
4 | nfrald.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
5 | 4 | nfnd 1862 | . . . 4 ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) |
6 | 2, 3, 5 | nfrald 3344 | . . 3 ⊢ (𝜑 → Ⅎ𝑥∀𝑦 ∈ 𝐴 ¬ 𝜓) |
7 | 6 | nfnd 1862 | . 2 ⊢ (𝜑 → Ⅎ𝑥 ¬ ∀𝑦 ∈ 𝐴 ¬ 𝜓) |
8 | 1, 7 | nfxfrd 1857 | 1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 Ⅎwnf 1786 Ⅎwnfc 2884 ∀wral 3061 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-13 2371 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-nf 1787 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ral 3062 df-rex 3071 |
This theorem is referenced by: nfrex 3347 nfiundg 47206 |
Copyright terms: Public domain | W3C validator |