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Mirrors > Home > MPE Home > Th. List > Mathboxes > reuxfr4d | Structured version Visualization version GIF version |
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Cf. reuxfrd 4893. (Contributed by Thierry Arnoux, 7-Apr-2017.) |
Ref | Expression |
---|---|
reuxfr4d.1 | ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) |
reuxfr4d.2 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) |
reuxfr4d.3 | ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reuxfr4d | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reuxfr4d.2 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ∃!𝑦 ∈ 𝐶 𝑥 = 𝐴) | |
2 | reurex 3160 | . . . . . 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 529 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓))) |
5 | r19.41v 3089 | . . . . . 6 ⊢ (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ (∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓)) | |
6 | reuxfr4d.3 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 ↔ 𝜒)) | |
7 | 6 | pm5.32da 673 | . . . . . . 7 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝜓) ↔ (𝑥 = 𝐴 ∧ 𝜒))) |
8 | 7 | rexbidv 3052 | . . . . . 6 ⊢ (𝜑 → (∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
9 | 5, 8 | syl5bbr 274 | . . . . 5 ⊢ (𝜑 → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → ((∃𝑦 ∈ 𝐶 𝑥 = 𝐴 ∧ 𝜓) ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
11 | 4, 10 | bitrd 268 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐵) → (𝜓 ↔ ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
12 | 11 | reubidva 3125 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒))) |
13 | reuxfr4d.1 | . . 3 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → 𝐴 ∈ 𝐵) | |
14 | reurmo 3161 | . . . 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 | reuxfr3d 29329 | . 2 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 ∃𝑦 ∈ 𝐶 (𝑥 = 𝐴 ∧ 𝜒) ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
17 | 12, 16 | bitrd 268 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐵 𝜓 ↔ ∃!𝑦 ∈ 𝐶 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∃wrex 2913 ∃!wreu 2914 ∃*wrmo 2915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-v 3202 |
This theorem is referenced by: rmoxfrdOLD 29332 rmoxfrd 29333 fcnvgreu 29472 |
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