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Mirrors > Home > MPE Home > Th. List > rmoim | 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 |
---|---|
rmoim | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ral 3071 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓))) | |
2 | imdistan 568 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) | |
3 | 2 | albii 1826 | . . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
4 | 1, 3 | bitri 274 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓))) |
5 | moim 2546 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
6 | df-rmo 3074 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
7 | df-rmo 3074 | . . 3 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
8 | 5, 6, 7 | 3imtr4g 296 | . 2 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐴 ∧ 𝜓)) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
9 | 4, 8 | sylbi 216 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (∃*𝑥 ∈ 𝐴 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∀wal 1540 ∈ wcel 2110 ∃*wmo 2540 ∀wral 3066 ∃*wrmo 3069 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-mo 2542 df-ral 3071 df-rmo 3074 |
This theorem is referenced by: rmoimia 3680 reuimrmo 3684 2rmorex 3693 2reurex 3699 disjss2 5047 catideu 17380 rinvmod 19406 frlmup4 21004 evlseu 21289 2ndcdisj 22603 2sqreulem1 26590 2sqreunnlem1 26593 poimirlem18 35789 poimirlem21 35792 |
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