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Mirrors > Home > MPE Home > Th. List > rmoanimALT | Structured version Visualization version GIF version |
Description: Alternate proof of rmoanim 3797, shorter but requiring ax-10 2141 and ax-11 2158. (Contributed by Alexander van der Vekens, 25-Jun-2017.) (New usage is discouraged.) (Proof modification is discouraged.) |
Ref | Expression |
---|---|
rmoanim.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
rmoanimALT | ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | impexp 454 | . . . . 5 ⊢ (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓 → 𝑥 = 𝑦))) | |
2 | 1 | ralbii 3081 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦))) |
3 | rmoanim.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
4 | 3 | r19.21 3129 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
5 | 2, 4 | bitri 278 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
6 | 5 | exbii 1855 | . 2 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
7 | nfv 1922 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝜓) | |
8 | 7 | rmo2 3790 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
9 | nfv 1922 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
10 | 9 | rmo2 3790 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) |
11 | 10 | imbi2i 339 | . . 3 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
12 | 19.37v 2000 | . . 3 ⊢ (∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | |
13 | 11, 12 | bitr4i 281 | . 2 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
14 | 6, 8, 13 | 3bitr4i 306 | 1 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∃wex 1787 Ⅎwnf 1791 ∀wral 3054 ∃*wrmo 3057 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-mo 2537 df-ral 3059 df-rmo 3062 |
This theorem is referenced by: (None) |
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