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Mirrors > Home > MPE Home > Th. List > rmo4f | Structured version Visualization version GIF version |
Description: Restricted "at most one" using implicit substitution. (Contributed by NM, 24-Oct-2006.) (Revised by Thierry Arnoux, 11-Oct-2016.) (Revised by Thierry Arnoux, 8-Mar-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.) |
Ref | Expression |
---|---|
rmo4f.1 | ⊢ Ⅎ𝑥𝐴 |
rmo4f.2 | ⊢ Ⅎ𝑦𝐴 |
rmo4f.3 | ⊢ Ⅎ𝑥𝜓 |
rmo4f.4 | ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
rmo4f | ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmo4f.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
2 | rmo4f.2 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
3 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦𝜑 | |
4 | 1, 2, 3 | rmo3f 3673 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦)) |
5 | rmo4f.3 | . . . . . 6 ⊢ Ⅎ𝑥𝜓 | |
6 | rmo4f.4 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | sbiev 2322 | . . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
8 | 7 | anbi2i 625 | . . . 4 ⊢ ((𝜑 ∧ [𝑦 / 𝑥]𝜑) ↔ (𝜑 ∧ 𝜓)) |
9 | 8 | imbi1i 353 | . . 3 ⊢ (((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
10 | 9 | 2ralbii 3134 | . 2 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ [𝑦 / 𝑥]𝜑) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
11 | 4, 10 | bitri 278 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 Ⅎwnf 1785 [wsb 2069 Ⅎwnfc 2936 ∀wral 3106 ∃*wrmo 3109 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-10 2142 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-clel 2870 df-nfc 2938 df-ral 3111 df-rmo 3114 |
This theorem is referenced by: 2sqreulem4 26038 disjorf 30342 funcnv5mpt 30431 |
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