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Mirrors > Home > MPE Home > Th. List > mobidv | Structured version Visualization version GIF version |
Description: Formula-building rule for the at-most-one quantifier (deduction form). (Contributed by Mario Carneiro, 7-Oct-2016.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.) |
Ref | Expression |
---|---|
mobidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
mobidv | ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mobidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | alrimiv 1886 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | mobi 2556 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝜑 → (∃*𝑥𝜓 ↔ ∃*𝑥𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∀wal 1505 ∃*wmo 2545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 |
This theorem depends on definitions: df-bi 199 df-an 388 df-ex 1743 df-mo 2547 |
This theorem is referenced by: moanimv 2653 rmoeq1 3342 mosubopt 5250 dffun6f 6196 funmo 6198 caovmo 7195 1stconst 7597 2ndconst 7598 brdom3 9742 brdom6disj 9746 imasaddfnlem 16651 imasvscafn 16660 hausflim 22287 hausflf 22303 cnextfun 22370 haustsms 22441 limcmo 24177 perfdvf 24198 phpreu 34317 alrmomodm 35059 funressnfv 42683 funressnmo 42687 |
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