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Mirrors > Home > MPE Home > Th. List > nfrmod | Structured version Visualization version GIF version |
Description: Deduction version of nfrmo 3427. Usage of this theorem is discouraged because it depends on ax-13 2367. (Contributed by NM, 17-Jun-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfrmod.1 | ⊢ Ⅎ𝑦𝜑 |
nfrmod.2 | ⊢ (𝜑 → Ⅎ𝑥𝐴) |
nfrmod.3 | ⊢ (𝜑 → Ⅎ𝑥𝜓) |
Ref | Expression |
---|---|
nfrmod | ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3373 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜓 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) | |
2 | nfrmod.1 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
3 | nfcvf 2929 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | 3 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝑦) |
5 | nfrmod.2 | . . . . . 6 ⊢ (𝜑 → Ⅎ𝑥𝐴) | |
6 | 5 | adantr 480 | . . . . 5 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝐴) |
7 | 4, 6 | nfeld 2911 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥 𝑦 ∈ 𝐴) |
8 | nfrmod.3 | . . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓) | |
9 | 8 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓) |
10 | 7, 9 | nfand 1893 | . . 3 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜓)) |
11 | 2, 10 | nfmod2 2548 | . 2 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜓)) |
12 | 1, 11 | nfxfrd 1849 | 1 ⊢ (𝜑 → Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∀wal 1532 Ⅎwnf 1778 ∈ wcel 2099 ∃*wmo 2528 Ⅎwnfc 2879 ∃*wrmo 3372 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-13 2367 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-mo 2530 df-cleq 2720 df-clel 2806 df-nfc 2881 df-rmo 3373 |
This theorem is referenced by: (None) |
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