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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 2136, ax-11 2151, and ax-12 2167. (Revised by Steven Nguyen, 30-Apr-2023.) |
Ref | Expression |
---|---|
reueq1 | ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2898 | . . . 4 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | anbi1d 629 | . . 3 ⊢ (𝐴 = 𝐵 → ((𝑥 ∈ 𝐴 ∧ 𝜑) ↔ (𝑥 ∈ 𝐵 ∧ 𝜑))) |
3 | 2 | eubidv 2665 | . 2 ⊢ (𝐴 = 𝐵 → (∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑) ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑))) |
4 | df-reu 3142 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ 𝜑)) | |
5 | df-reu 3142 | . 2 ⊢ (∃!𝑥 ∈ 𝐵 𝜑 ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ 𝜑)) | |
6 | 3, 4, 5 | 3bitr4g 315 | 1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ 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 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-ex 1772 df-mo 2615 df-eu 2647 df-cleq 2811 df-clel 2890 df-reu 3142 |
This theorem is referenced by: reueqd 3417 lubfval 17576 glbfval 17589 uspgredg2vlem 26932 uspgredg2v 26933 isfrgr 27966 frgr1v 27977 nfrgr2v 27978 frgr3v 27981 1vwmgr 27982 3vfriswmgr 27984 isplig 28180 hdmap14lem4a 38887 hdmap14lem15 38898 |
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