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| Mirrors > Home > MPE Home > Th. List > rexprg | Structured version Visualization version GIF version | ||
| Description: Convert a restricted existential quantification over a pair to a disjunction. (Contributed by NM, 17-Sep-2011.) (Revised by Mario Carneiro, 23-Apr-2015.) Avoid ax-10 2141, ax-12 2177. (Revised by GG, 30-Sep-2024.) |
| Ref | Expression |
|---|---|
| ralprg.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| ralprg.2 | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| rexprg | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ralprg.1 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 2 | 1 | notbid 318 | . . 3 ⊢ (𝑥 = 𝐴 → (¬ 𝜑 ↔ ¬ 𝜓)) |
| 3 | ralprg.2 | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
| 4 | 3 | notbid 318 | . . 3 ⊢ (𝑥 = 𝐵 → (¬ 𝜑 ↔ ¬ 𝜒)) |
| 5 | 2, 4 | ralprg 4696 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒))) |
| 6 | ralnex 3072 | . . . 4 ⊢ (∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ ¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑) | |
| 7 | pm4.56 991 | . . . 4 ⊢ ((¬ 𝜓 ∧ ¬ 𝜒) ↔ ¬ (𝜓 ∨ 𝜒)) | |
| 8 | 6, 7 | bibi12i 339 | . . 3 ⊢ ((∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)) ↔ (¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ¬ (𝜓 ∨ 𝜒))) |
| 9 | notbi 319 | . . 3 ⊢ ((∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒)) ↔ (¬ ∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ¬ (𝜓 ∨ 𝜒))) | |
| 10 | 8, 9 | sylbb2 238 | . 2 ⊢ ((∀𝑥 ∈ {𝐴, 𝐵} ¬ 𝜑 ↔ (¬ 𝜓 ∧ ¬ 𝜒)) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
| 11 | 5, 10 | syl 17 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 {cpr 4628 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-v 3482 df-un 3956 df-sn 4627 df-pr 4629 |
| This theorem is referenced by: rextpg 4699 rexpr 4701 reurexprg 4704 fr2nr 5662 sgrp2nmndlem5 18942 nb3grprlem2 29398 nfrgr2v 30291 3vfriswmgrlem 30296 brfvrcld 43704 rnmptpr 45182 ldepspr 48390 zlmodzxzldeplem4 48420 |
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