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Mirrors > Home > MPE Home > Th. List > mobid | Structured version Visualization version GIF version |
Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by NM, 8-Mar-1995.) Remove dependency on ax-10 2142, ax-11 2159, ax-13 2372. (Revised by BJ, 14-Oct-2022.) (Proof shortened by Wolf Lammen, 18-Feb-2023.) |
Ref | Expression |
---|---|
mobid.1 | ⊢ Ⅎ𝑥𝜑 |
mobid.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mobid | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mobid.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | mobid.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimi 2212 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | mobi 2547 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∀wal 1541 Ⅎwnf 1791 ∃*wmo 2538 |
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-12 2176 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1788 df-nf 1792 df-mo 2540 |
This theorem is referenced by: moanim 2622 rmobida 3316 rmoeq1f 3324 funcnvmpt 30748 |
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