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Mirrors > Home > MPE Home > Th. List > 2reu2rex | Structured version Visualization version GIF version |
Description: Double restricted existential uniqueness, analogous to 2eu2ex 2639. (Contributed by Alexander van der Vekens, 25-Jun-2017.) |
Ref | Expression |
---|---|
2reu2rex | ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reurex 3380 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑) | |
2 | reurex 3380 | . . 3 ⊢ (∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑦 ∈ 𝐵 𝜑) | |
3 | 2 | reximi 3084 | . 2 ⊢ (∃𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
4 | 1, 3 | syl 17 | 1 ⊢ (∃!𝑥 ∈ 𝐴 ∃!𝑦 ∈ 𝐵 𝜑 → ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∃wrex 3070 ∃!wreu 3374 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1782 df-eu 2563 df-rex 3071 df-rmo 3376 df-reu 3377 |
This theorem is referenced by: 2reu1 3890 2sqreultblem 26940 2sqreunnltblem 26943 |
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