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Mirrors > Home > MPE Home > Th. List > reuxfr1d | Structured version Visualization version GIF version |
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfr1ds 3690. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
Ref | Expression |
---|---|
reuxfr1d.1 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
reuxfr1d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
reuxfr1d.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reuxfr1d | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuxfr1d.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
2 | reurex 3376 | . . . . . 6 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
3 | 1, 2 | syl 17 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃𝑦 ∈ 𝐶 𝑥 = 𝐴) |
4 | 3 | biantrurd 536 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓))) |
5 | r19.41v 3300 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓)) | |
6 | reuxfr1d.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
7 | 6 | pm5.32da 582 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒))) |
8 | 7 | rexbidv 3256 | . . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
9 | 5, 8 | bitr3id 288 | . . . . 5 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
10 | 9 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
11 | 4, 10 | bitrd 282 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
12 | 11 | reubidva 3341 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
13 | reuxfr1d.1 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) | |
14 | reurmo 3378 | . . . 4 ⊢ (∃!𝑦 ∈ 𝐶 𝑥 = 𝐴 → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
15 | 1, 14 | syl 17 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃*𝑦 ∈ 𝐶 𝑥 = 𝐴) |
16 | 13, 15 | reuxfrd 3687 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒) ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
17 | 12, 16 | bitrd 282 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∃wrex 3107 ∃!wreu 3108 ∃*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-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 |
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-eu 2629 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 |
This theorem is referenced by: reuxfr1ds 3690 rmoxfrd 30264 fcnvgreu 30436 reuf1odnf 43663 reuf1od 43664 |
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