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Mirrors > Home > MPE Home > Th. List > reueq1 | Structured version Visualization version GIF version |
Description: Equality theorem for restricted unique existential quantifier. (Contributed by NM, 5-Apr-2004.) Remove usage of ax-10 2130, ax-11 2147, and ax-12 2167. (Revised by Steven Nguyen, 30-Apr-2023.) Avoid ax-8 2101. (Revised by Wolf Lammen, 12-Mar-2025.) |
Ref | Expression |
---|---|
reueq1 | ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexeq 3317 | . . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑥 ∈ 𝐵 𝜑)) | |
2 | rmoeq1 3410 | . . 3 ⊢ (𝐴 = 𝐵 → (∃*𝑥 ∈ 𝐴 𝜑 ↔ ∃*𝑥 ∈ 𝐵 𝜑)) | |
3 | 1, 2 | anbi12d 631 | . 2 ⊢ (𝐴 = 𝐵 → ((∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑) ↔ (∃𝑥 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑))) |
4 | reu5 3374 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ (∃𝑥 ∈ 𝐴 𝜑 ∧ ∃*𝑥 ∈ 𝐴 𝜑)) | |
5 | reu5 3374 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ (∃𝑥 ∈ 𝐵 𝜑 ∧ ∃*𝑥 ∈ 𝐵 𝜑)) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∃wrex 3066 ∃!wreu 3370 ∃*wrmo 3371 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1775 df-mo 2530 df-eu 2559 df-cleq 2720 df-rex 3067 df-rmo 3372 df-reu 3373 |
This theorem is referenced by: reueqd 3415 lubfval 18335 glbfval 18348 uspgredg2vlem 29029 uspgredg2v 29030 isfrgr 30063 frgr1v 30074 nfrgr2v 30075 frgr3v 30078 1vwmgr 30079 3vfriswmgr 30081 isplig 30279 hdmap14lem4a 41338 hdmap14lem15 41349 |
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