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Mirrors > Home > MPE Home > Th. List > eqreu | Structured version Visualization version GIF version |
Description: A condition which implies existential uniqueness. (Contributed by Mario Carneiro, 2-Oct-2015.) |
Ref | Expression |
---|---|
eqreu.1 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
eqreu | ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralbiim 3141 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑))) | |
2 | eqreu.1 | . . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 2 | ceqsralv 3480 | . . . . . 6 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑) ↔ 𝜓)) |
4 | 3 | anbi2d 631 | . . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ ∀𝑥 ∈ 𝐴 (𝑥 = 𝐵 → 𝜑)) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓))) |
5 | 1, 4 | syl5bb 286 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓))) |
6 | reu6i 3667 | . . . . 5 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) | |
7 | 6 | ex 416 | . . . 4 ⊢ (𝐵 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 (𝜑 ↔ 𝑥 = 𝐵) → ∃!𝑥 ∈ 𝐴 𝜑)) |
8 | 5, 7 | sylbird 263 | . . 3 ⊢ (𝐵 ∈ 𝐴 → ((∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑)) |
9 | 8 | 3impib 1113 | . 2 ⊢ ((𝐵 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵) ∧ 𝜓) → ∃!𝑥 ∈ 𝐴 𝜑) |
10 | 9 | 3com23 1123 | 1 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝜓 ∧ ∀𝑥 ∈ 𝐴 (𝜑 → 𝑥 = 𝐵)) → ∃!𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ∃!wreu 3108 |
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-12 2175 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-cleq 2791 df-clel 2870 df-ral 3111 df-rex 3112 df-reu 3113 |
This theorem is referenced by: uzwo3 12331 frmdup3 18024 frgpup3 18896 neiptopreu 21738 ufileu 22524 mirreu 26458 lmireu 26584 opreu2reuALT 30247 symgfcoeu 30776 |
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