Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > rmoimi2 | 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 |
---|---|
rmoimi2.1 | ⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) |
Ref | Expression |
---|---|
rmoimi2 | ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rmoimi2.1 | . . 3 ⊢ ∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) | |
2 | moim 2519 | . . 3 ⊢ (∀𝑥((𝑥 ∈ 𝐴 ∧ 𝜑) → (𝑥 ∈ 𝐵 ∧ 𝜓)) → (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑))) | |
3 | 1, 2 | ax-mp 5 | . 2 ⊢ (∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓) → ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) |
4 | df-rmo 2920 | . 2 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 ↔ ∃*𝑥(𝑥 ∈ 𝐵 ∧ 𝜓)) | |
5 | df-rmo 2920 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
6 | 3, 4, 5 | 3imtr4i 281 | 1 ⊢ (∃*𝑥 ∈ 𝐵 𝜓 → ∃*𝑥 ∈ 𝐴 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∃*wmo 2471 ∃*wrmo 2915 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-10 2019 ax-12 2047 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-ex 1705 df-nf 1710 df-eu 2474 df-mo 2475 df-rmo 2920 |
This theorem is referenced by: disjin 29399 disjin2 29400 |
Copyright terms: Public domain | W3C validator |