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Mathbox for Stefan O'Rear |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > moxfr | Structured version Visualization version GIF version |
Description: Transfer at-most-one between related expressions. (Contributed by Stefan O'Rear, 12-Feb-2015.) |
Ref | Expression |
---|---|
moxfr.a | ⊢ 𝐴 ∈ V |
moxfr.b | ⊢ ∃!𝑦 𝑥 = 𝐴 |
moxfr.c | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
moxfr | ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | moxfr.a | . . . . . 6 ⊢ 𝐴 ∈ V | |
2 | 1 | a1i 11 | . . . . 5 ⊢ (𝑦 ∈ V → 𝐴 ∈ V) |
3 | moxfr.b | . . . . . . . 8 ⊢ ∃!𝑦 𝑥 = 𝐴 | |
4 | euex 2597 | . . . . . . . 8 ⊢ (∃!𝑦 𝑥 = 𝐴 → ∃𝑦 𝑥 = 𝐴) | |
5 | 3, 4 | ax-mp 5 | . . . . . . 7 ⊢ ∃𝑦 𝑥 = 𝐴 |
6 | rexv 3422 | . . . . . . 7 ⊢ (∃𝑦 ∈ V 𝑥 = 𝐴 ↔ ∃𝑦 𝑥 = 𝐴) | |
7 | 5, 6 | mpbir 223 | . . . . . 6 ⊢ ∃𝑦 ∈ V 𝑥 = 𝐴 |
8 | 7 | a1i 11 | . . . . 5 ⊢ (𝑥 ∈ V → ∃𝑦 ∈ V 𝑥 = 𝐴) |
9 | moxfr.c | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
10 | 2, 8, 9 | rexxfr 5128 | . . . 4 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑦 ∈ V 𝜓) |
11 | rexv 3422 | . . . 4 ⊢ (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑) | |
12 | rexv 3422 | . . . 4 ⊢ (∃𝑦 ∈ V 𝜓 ↔ ∃𝑦𝜓) | |
13 | 10, 11, 12 | 3bitr3i 293 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓) |
14 | 1, 3, 9 | euxfr 3604 | . . 3 ⊢ (∃!𝑥𝜑 ↔ ∃!𝑦𝜓) |
15 | 13, 14 | imbi12i 342 | . 2 ⊢ ((∃𝑥𝜑 → ∃!𝑥𝜑) ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) |
16 | moeu 2603 | . 2 ⊢ (∃*𝑥𝜑 ↔ (∃𝑥𝜑 → ∃!𝑥𝜑)) | |
17 | moeu 2603 | . 2 ⊢ (∃*𝑦𝜓 ↔ (∃𝑦𝜓 → ∃!𝑦𝜓)) | |
18 | 15, 16, 17 | 3bitr4i 295 | 1 ⊢ (∃*𝑥𝜑 ↔ ∃*𝑦𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1601 ∃wex 1823 ∈ wcel 2107 ∃*wmo 2549 ∃!weu 2586 ∃wrex 3091 Vcvv 3398 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-v 3400 |
This theorem is referenced by: (None) |
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