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Mirrors > Home > MPE Home > Th. List > nfrmo | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for restricted uniqueness. Usage of this theorem is discouraged because it depends on ax-13 2369. Use the weaker nfrmow 3407 when possible. (Contributed by NM, 16-Jun-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nfrmo.1 | ⊢ Ⅎ𝑥𝐴 |
nfrmo.2 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
nfrmo | ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rmo 3374 | . 2 ⊢ (∃*𝑦 ∈ 𝐴 𝜑 ↔ ∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) | |
2 | nftru 1804 | . . . 4 ⊢ Ⅎ𝑦⊤ | |
3 | nfcvf 2930 | . . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦) | |
4 | nfrmo.1 | . . . . . . . 8 ⊢ Ⅎ𝑥𝐴 | |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐴) |
6 | 3, 5 | nfeld 2912 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥 𝑦 ∈ 𝐴) |
7 | nfrmo.2 | . . . . . . 7 ⊢ Ⅎ𝑥𝜑 | |
8 | 7 | a1i 11 | . . . . . 6 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝜑) |
9 | 6, 8 | nfand 1898 | . . . . 5 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
10 | 9 | adantl 480 | . . . 4 ⊢ ((⊤ ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝜑)) |
11 | 2, 10 | nfmod2 2550 | . . 3 ⊢ (⊤ → Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑)) |
12 | 11 | mptru 1546 | . 2 ⊢ Ⅎ𝑥∃*𝑦(𝑦 ∈ 𝐴 ∧ 𝜑) |
13 | 1, 12 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥∃*𝑦 ∈ 𝐴 𝜑 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 394 ∀wal 1537 ⊤wtru 1540 Ⅎwnf 1783 ∈ wcel 2104 ∃*wmo 2530 Ⅎwnfc 2881 ∃*wrmo 3373 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-13 2369 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-tru 1542 df-ex 1780 df-nf 1784 df-mo 2532 df-cleq 2722 df-clel 2808 df-nfc 2883 df-rmo 3374 |
This theorem is referenced by: (None) |
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