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| Mirrors > Home > MPE Home > Th. List > rmoimi | Structured version Visualization version GIF version | ||
| Description: Restricted "at most one" is preserved through implication (note wff reversal). (Contributed by Alexander van der Vekens, 17-Jun-2017.) | 
| Ref | Expression | 
|---|---|
| rmoimi.1 | ⊢ (𝜑 → 𝜓) | 
| Ref | Expression | 
|---|---|
| rmoimi | ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rmoimi.1 | . . 3 ⊢ (𝜑 → 𝜓) | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | 
| 3 | 2 | rmoimia 3747 | 1 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2108 ∃*wrmo 3379 | 
| 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 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-mo 2540 df-ral 3062 df-rmo 3380 | 
| This theorem is referenced by: 2rexreu 3768 2sqreunnlem1 27493 disjin 32599 disjin2 32600 addinvcom 42461 | 
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