Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 2587 | . 2 ⊢ (𝜑 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒))) |
4 | df-reu 3073 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
5 | df-reu 3073 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜒 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜒)) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝜑 → (∃!𝑥 ∈ 𝐴 𝜓 ↔ ∃!𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2107 ∃!weu 2569 ∃!wreu 3067 |
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 397 df-ex 1783 df-mo 2541 df-eu 2570 df-reu 3073 |
This theorem is referenced by: reubidv 3324 reuxfrd 3684 reuxfr1d 3686 exfo 6990 f1ofveu 7279 zmax 12694 zbtwnre 12695 rebtwnz 12696 icoshftf1o 13215 divalgb 16122 1arith2 16638 ply1divalg2 25312 addsq2reu 26597 addsqn2reu 26598 addsqrexnreu 26599 2sqreultlem 26604 2sqreunnltlem 26607 frgr2wwlkeu 28700 numclwwlk2lem1 28749 numclwlk2lem2f1o 28752 pjhtheu2 29787 reuxfrdf 30848 xrsclat 31298 xrmulc1cn 31889 poimirlem25 35811 hdmap14lem14 39902 prproropreud 44972 quad1 45083 requad1 45085 requad2 45086 itscnhlinecirc02p 46142 |
Copyright terms: Public domain | W3C validator |