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Mirrors > Home > MPE Home > Th. List > rexcomf | Structured version Visualization version GIF version |
Description: Commutation of restricted existential quantifiers. For a version based on fewer axioms see rexcom 3281. (Contributed by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
ralcomf.1 | ⊢ Ⅎ𝑦𝐴 |
ralcomf.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
rexcomf | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ancom 460 | . . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ↔ (𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴)) | |
2 | 1 | anbi1i 623 | . . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
3 | 2 | 2exbii 1843 | . . 3 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
4 | excom 2154 | . . 3 ⊢ (∃𝑥∃𝑦((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) | |
5 | 3, 4 | bitri 275 | . 2 ⊢ (∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
6 | ralcomf.1 | . . 3 ⊢ Ⅎ𝑦𝐴 | |
7 | 6 | r2exf 3273 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥∃𝑦((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑)) |
8 | ralcomf.2 | . . 3 ⊢ Ⅎ𝑥𝐵 | |
9 | 8 | r2exf 3273 | . 2 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦∃𝑥((𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐴) ∧ 𝜑)) |
10 | 5, 7, 9 | 3bitr4i 303 | 1 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 395 ∃wex 1773 ∈ wcel 2098 Ⅎwnfc 2877 ∃wrex 3064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-11 2146 ax-12 2163 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-nf 1778 df-clel 2804 df-nfc 2879 df-ral 3056 df-rex 3065 |
This theorem is referenced by: rexcom4f 32215 |
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