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Mirrors > Home > MPE Home > Th. List > Mathboxes > mobidvALT | Structured version Visualization version GIF version |
Description: Alternate proof of mobidv 2542 directly from its analogues albidv 1922 and exbidv 1923, using deduction style. Note the proof structure, similar to mobi 2540. (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. Remove dependency on ax-6 1970, ax-7 2010, ax-12 2170 by adapting proof of mobid 2543. (Revised by BJ, 26-Sep-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mobidvALT.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mobidvALT | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mobidvALT.1 | . . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | imbi1d 340 | . . . 4 ⊢ (𝜑 → ((𝜓 → 𝑥 = 𝑦) ↔ (𝜒 → 𝑥 = 𝑦))) |
3 | 2 | albidv 1922 | . . 3 ⊢ (𝜑 → (∀𝑥(𝜓 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜒 → 𝑥 = 𝑦))) |
4 | 3 | exbidv 1923 | . 2 ⊢ (𝜑 → (∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦) ↔ ∃𝑦∀𝑥(𝜒 → 𝑥 = 𝑦))) |
5 | df-mo 2533 | . 2 ⊢ (∃*𝑥𝜓 ↔ ∃𝑦∀𝑥(𝜓 → 𝑥 = 𝑦)) | |
6 | df-mo 2533 | . 2 ⊢ (∃*𝑥𝜒 ↔ ∃𝑦∀𝑥(𝜒 → 𝑥 = 𝑦)) | |
7 | 4, 5, 6 | 3bitr4g 313 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1538 ∃wex 1780 ∃*wmo 2531 |
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 1912 |
This theorem depends on definitions: df-bi 206 df-ex 1781 df-mo 2533 |
This theorem is referenced by: (None) |
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