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Mirrors > Home > MPE Home > Th. List > eubidv | Structured version Visualization version GIF version |
Description: Formula-building rule for unique existential quantifier (deduction form). (Contributed by NM, 9-Jul-1994.) Reduce axiom dependencies and shorten proof. (Revised by BJ, 7-Oct-2022.) |
Ref | Expression |
---|---|
eubidv.1 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
eubidv | ⊢ (𝜑 → (∃!𝑥𝜓 ↔ ∃!𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eubidv.1 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
2 | 1 | alrimiv 1931 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
3 | eubi 2579 | . 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 205 ∀wal 1540 ∃!weu 2563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-mo 2535 df-eu 2564 |
This theorem is referenced by: euorv 2609 euanv 2621 reubidva 3393 reueubd 3396 reueq1OLD 3418 eueq2 3707 eueq3 3708 moeq3 3709 reusv2lem2 5398 reusv2lem5 5401 reuhypd 5418 feu 6768 dff3 7102 dff4 7103 omxpenlem 9073 dfac5lem5 10122 dfac5 10123 kmlem2 10146 kmlem12 10156 kmlem13 10157 initoval 17943 termoval 17944 isinito 17946 istermo 17947 initoid 17951 termoid 17952 initoeu1 17961 initoeu2 17966 termoeu1 17968 upxp 23127 edgnbusgreu 28624 nbusgredgeu0 28625 frgrncvvdeqlem2 29553 bnj852 33932 bnj1489 34067 funpartfv 34917 fsuppind 41162 fourierdlem36 44859 aiotaval 45803 eu2ndop1stv 45833 dfdfat2 45836 tz6.12-afv 45881 tz6.12-afv2 45948 dfatcolem 45963 prprsprreu 46187 prprreueq 46188 setrec2lem1 47738 |
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