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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 2139, ax-11 2154, ax-13 2384. (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 2206 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | mobi 2624 | . 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 208 ∀wal 1529 Ⅎwnf 1778 ∃*wmo 2614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-12 2170 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1775 df-nf 1779 df-mo 2616 |
This theorem is referenced by: moanim 2699 rmobida 3391 rmoeq1f 3399 funcnvmpt 30404 |
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