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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 581 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑥 ∈ 𝐴 ∧ 𝜒))) |
3 | 2 | eubidv 2668 | . 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | df-reu 3145 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | df-reu 3145 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
6 | 3, 4, 5 | 3bitr4g 316 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∈ wcel 2110 ∃!weu 2649 ∃!wreu 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-mo 2618 df-eu 2650 df-reu 3145 |
This theorem is referenced by: reubidv 3389 reuxfrd 3738 reuxfr1d 3740 exfo 6865 f1ofveu 7145 zmax 12339 zbtwnre 12340 rebtwnz 12341 icoshftf1o 12854 divalgb 15749 1arith2 16258 ply1divalg2 24726 addsq2reu 26010 addsqn2reu 26011 addsqrexnreu 26012 2sqreultlem 26017 2sqreunnltlem 26020 frgr2wwlkeu 28100 numclwwlk2lem1 28149 numclwlk2lem2f1o 28152 pjhtheu2 29187 reuxfrdf 30249 xrsclat 30662 xrmulc1cn 31168 poimirlem25 34911 hdmap14lem14 39011 prproropreud 43665 quad1 43779 requad1 43781 requad2 43782 itscnhlinecirc02p 44766 |
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