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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 3802 | . 2 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓)) → (𝐵 = 𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝜒))) |
4 | 3 | biimp3ar 1472 | 1 ⊢ ((∃*𝑥 ∈ 𝐴 𝜑 ∧ (𝐵 ∈ 𝐴 ∧ 𝜓) ∧ (𝐶 ∈ 𝐴 ∧ 𝜒)) → 𝐵 = 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2110 ∃*wrmo 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-clab 2715 df-cleq 2729 df-clel 2816 df-rmo 3069 df-v 3410 |
This theorem is referenced by: eqsqrtd 14931 efgred2 19143 0frgp 19169 frgpnabllem2 19259 frgpcyg 20538 cdleme0moN 37976 proot1mul 40727 |
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