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Mirrors > Home > MPE Home > Th. List > reubidva | Structured version Visualization version GIF version |
Description: Formula-building rule for restricted existential quantifier (deduction form). (Contributed by NM, 13-Nov-2004.) Reduce axiom usage. (Revised by Wolf Lammen, 14-Jan-2023.) |
Ref | Expression |
---|---|
reubidva.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
reubidva | ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | reubidva.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 ↔ 𝜒)) | |
2 | 1 | pm5.32da 579 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
3 | 2 | eubidv 2665 | . 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | df-reu 3142 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | df-reu 3142 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
6 | 3, 4, 5 | 3bitr4g 315 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∈ wcel 2105 ∃!weu 2646 ∃!wreu 3137 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-mo 2615 df-eu 2647 df-reu 3142 |
This theorem is referenced by: reubidv 3387 reuxfrd 3736 reuxfr1d 3738 exfo 6863 f1ofveu 7140 zmax 12333 zbtwnre 12334 rebtwnz 12335 icoshftf1o 12848 divalgb 15743 1arith2 16252 ply1divalg2 24659 addsq2reu 25943 addsqn2reu 25944 addsqrexnreu 25945 2sqreultlem 25950 2sqreunnltlem 25953 frgr2wwlkeu 28033 numclwwlk2lem1 28082 numclwlk2lem2f1o 28085 pjhtheu2 29120 reuxfrdf 30182 xrsclat 30594 xrmulc1cn 31072 poimirlem25 34798 hdmap14lem14 38897 prproropreud 43548 quad1 43662 requad1 43664 requad2 43665 itscnhlinecirc02p 44700 |
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