![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > rexcomOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rexcom 3281 as of 8-Dec-2024. (Contributed by NM, 19-Nov-1995.) (Revised by Mario Carneiro, 14-Oct-2016.) (Proof shortened by BJ, 26-Aug-2023.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rexcomOLD | ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-rex 3065 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) | |
2 | 1 | rexbii 3088 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) |
3 | rexcom4 3279 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑)) | |
4 | r19.42v 3184 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ (𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)) | |
5 | 4 | exbii 1842 | . . 3 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)) |
6 | df-rex 3065 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)) | |
7 | 5, 6 | bitr4i 278 | . 2 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝜑) |
8 | 2, 3, 7 | 3bitri 297 | 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 ∃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-11 2146 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-rex 3065 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |