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Mirrors > Home > MPE Home > Th. List > rexprgf | Structured version Visualization version GIF version |
Description: Convert a restricted existential quantification over a pair to a disjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 2-Apr-2023.) |
Ref | Expression |
---|---|
ralprgf.1 | ⊢ Ⅎ𝑥𝜓 |
ralprgf.2 | ⊢ Ⅎ𝑥𝜒 |
ralprgf.a | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralprgf.b | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
rexprgf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4563 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | rexeqi 3414 | . . 3 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑) |
3 | rexun 4165 | . . 3 ⊢ (∃𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑)) | |
4 | 2, 3 | bitri 277 | . 2 ⊢ (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑)) |
5 | ralprgf.1 | . . . . 5 ⊢ Ⅎ𝑥𝜓 | |
6 | ralprgf.a | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | rexsngf 4603 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
8 | 7 | orbi1d 913 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑))) |
9 | ralprgf.2 | . . . . 5 ⊢ Ⅎ𝑥𝜒 | |
10 | ralprgf.b | . . . . 5 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
11 | 9, 10 | rexsngf 4603 | . . . 4 ⊢ (𝐵 ∈ 𝑊 → (∃𝑥 ∈ {𝐵}𝜑 ↔ 𝜒)) |
12 | 11 | orbi2d 912 | . . 3 ⊢ (𝐵 ∈ 𝑊 → ((𝜓 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ 𝜒))) |
13 | 8, 12 | sylan9bb 512 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∃𝑥 ∈ {𝐴}𝜑 ∨ ∃𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∨ 𝜒))) |
14 | 4, 13 | syl5bb 285 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∨ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ∃wrex 3139 ∪ cun 3933 {csn 4560 {cpr 4562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-v 3496 df-sbc 3772 df-un 3940 df-sn 4561 df-pr 4563 |
This theorem is referenced by: rexprg 4626 reuprg0 4631 |
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