Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4573 | . . . 4 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
2 | 1 | raleqi 3416 | . . 3 ⊢ (∀𝑥 ∈ {𝐴, 𝐵}𝜑 ↔ ∀𝑥 ∈ ({𝐴} ∪ {𝐵})𝜑) |
3 | ralunb 4170 | . . 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 4614 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ {𝐴}𝜑 ↔ 𝜓)) |
8 | ralprgf.2 | . . . 4 ⊢ Ⅎ𝑥𝜒 | |
9 | ralprgf.b | . . . 4 ⊢ (𝑥 = 𝐵 → (𝜑 ↔ 𝜒)) | |
10 | 8, 9 | ralsngf 4614 | . . 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 1536 Ⅎwnf 1783 ∈ wcel 2113 ∀wral 3141 ∪ cun 3937 {csn 4570 {cpr 4572 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-v 3499 df-sbc 3776 df-un 3944 df-sn 4571 df-pr 4573 |
This theorem is referenced by: ralprg 4635 |
Copyright terms: Public domain | W3C validator |