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Mirrors > Home > MPE Home > Th. List > rmoi | Structured version Visualization version GIF version |
Description: Consequence of "at most one", using implicit substitution. (Contributed by NM, 4-Nov-2012.) (Revised by NM, 16-Jun-2017.) |
Ref | Expression |
---|---|
rmoi.b | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
rmoi.c | ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
rmoi | ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmoi.b | . . 3 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
2 | rmoi.c | . . 3 ⊢ (𝑥 = 𝐶 → (𝜑 ↔ 𝜒)) | |
3 | 1, 2 | rmob 3495 | . 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))) |
4 | 3 | biimp3ar 1425 | 1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∃*wrmo 2899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rmo 2904 df-v 3175 |
This theorem is referenced by: eqsqrtd 13955 efgred2 17989 0frgp 18015 frgpnabllem2 18100 frgpcyg 19741 cdleme0moN 34530 proot1mul 36796 |
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