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Mirrors > Home > MPE Home > Th. List > rexcomOLD | Structured version Visualization version GIF version |
Description: Obsolete version of rexcom 3272 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 3071 | . . 3 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) | |
2 | 1 | rexbii 3094 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)) |
3 | rexcom4 3270 | . 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 1851 | . . 3 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ ∃𝑥 ∈ 𝐴 𝜑)) |
6 | df-rex 3071 | . . 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 397 ∃wex 1782 ∈ wcel 2107 ∃wrex 3070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-11 2155 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-rex 3071 |
This theorem is referenced by: (None) |
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