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Mirrors > Home > MPE Home > Th. List > rmoanimALT | Structured version Visualization version GIF version |
Description: Alternate proof of rmoanim 3878, shorter but requiring ax-10 2145 and ax-11 2161. (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 453 | . . . . 5 ⊢ (((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → (𝜓 → 𝑥 = 𝑦))) | |
2 | 1 | ralbii 3165 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦))) |
3 | rmoanim.1 | . . . . 5 ⊢ Ⅎ𝑥𝜑 | |
4 | 3 | r19.21 3215 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
5 | 2, 4 | bitri 277 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
6 | 5 | exbii 1848 | . 2 ⊢ (∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
7 | nfv 1915 | . . 3 ⊢ Ⅎ𝑦(𝜑 ∧ 𝜓) | |
8 | 7 | rmo2 3870 | . 2 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ ∃𝑦∀𝑥 ∈ 𝐴 ((𝜑 ∧ 𝜓) → 𝑥 = 𝑦)) |
9 | nfv 1915 | . . . . 5 ⊢ Ⅎ𝑦𝜓 | |
10 | 9 | rmo2 3870 | . . . 4 ⊢ (∃*𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) |
11 | 10 | imbi2i 338 | . . 3 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
12 | 19.37v 1998 | . . 3 ⊢ (∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦)) ↔ (𝜑 → ∃𝑦∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) | |
13 | 11, 12 | bitr4i 280 | . 2 ⊢ ((𝜑 → ∃*𝑥 ∈ 𝐴 𝜓) ↔ ∃𝑦(𝜑 → ∀𝑥 ∈ 𝐴 (𝜓 → 𝑥 = 𝑦))) |
14 | 6, 8, 13 | 3bitr4i 305 | 1 ⊢ (∃*𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 → ∃*𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∃wex 1780 Ⅎwnf 1784 ∀wral 3138 ∃*wrmo 3141 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-10 2145 ax-11 2161 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-mo 2622 df-ral 3143 df-rmo 3146 |
This theorem is referenced by: (None) |
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