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Mirrors > Home > MPE Home > Th. List > ralprgf | Structured version Visualization version GIF version |
Description: Convert a restricted universal quantification over a pair to a conjunction, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 17-Sep-2011.) (Revised by AV, 8-Apr-2023.) |
Ref | Expression |
---|---|
ralprgf.1 | ⊢ Ⅎ𝑥𝜓 |
ralprgf.2 | ⊢ Ⅎ𝑥𝜒 |
ralprgf.a | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ralprgf.b | ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) |
Ref | Expression |
---|---|
ralprgf | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 4570 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | raleqi 3413 | . . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑) |
3 | ralunb 4167 | . . 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 | . . . 4 ⊢ Ⅎ𝑥𝜓 | |
6 | ralprgf.a | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
7 | 5, 6 | ralsngf 4611 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
8 | ralprgf.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
9 | ralprgf.b | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
10 | 8, 9 | ralsngf 4611 | . . 3 ⊢ (𝐵 ∈ 𝑊 → (∀𝑥 ∈ {𝐵}𝜑 ↔ 𝜒)) |
11 | 7, 10 | bi2anan9 637 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ((∀𝑥 ∈ {𝐴}𝜑 ∧ ∀𝑥 ∈ {𝐵}𝜑) ↔ (𝜓 ∧ 𝜒))) |
12 | 4, 11 | 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 = wceq 1537 Ⅎwnf 1784 ∈ wcel 2114 ∀wral 3138 ∪ cun 3934 {csn 4567 {cpr 4569 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-v 3496 df-sbc 3773 df-un 3941 df-sn 4568 df-pr 4570 |
This theorem is referenced by: ralprg 4632 |
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