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Mirrors > Home > MPE Home > Th. List > euxfr | Structured version Visualization version GIF version |
Description: Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Usage of this theorem is discouraged because it depends on ax-13 2379. Use the weaker euxfrw 3660 when possible. (Contributed by NM, 14-Nov-2004.) (New usage is discouraged.) |
Ref | Expression |
---|---|
euxfr.1 | ⊢ 𝐴 ∈ V |
euxfr.2 | ⊢ ∃!𝑦 𝑥 = 𝐴 |
euxfr.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
euxfr | ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | euxfr.2 | . . . . . 6 ⊢ ∃!𝑦 𝑥 = 𝐴 | |
2 | euex 2637 | . . . . . 6 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ ∃𝑦 𝑥 = 𝐴 |
4 | 3 | biantrur 534 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑)) |
5 | 19.41v 1950 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ (∃𝑦 𝑥 = 𝐴 ∧ 𝜑)) | |
6 | euxfr.3 | . . . . . 6 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 6 | pm5.32i 578 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ 𝜓)) |
8 | 7 | exbii 1849 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
9 | 4, 5, 8 | 3bitr2i 302 | . . 3 ⊢ (𝜑 ↔ ∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
10 | 9 | eubii 2645 | . 2 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓)) |
11 | euxfr.1 | . . 3 ⊢ 𝐴 ∈ V | |
12 | 1 | eumoi 2639 | . . 3 ⊢ ∃*𝑦 𝑥 = 𝐴 |
13 | 11, 12 | euxfr2 3661 | . 2 ⊢ (∃!𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝜓) ↔ ∃!𝑦𝜓) |
14 | 10, 13 | bitri 278 | 1 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 ∃!weu 2628 Vcvv 3441 |
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-13 2379 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 |
This theorem is referenced by: (None) |
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