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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.) |
Ref | Expression |
---|---|
reueq1 | ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2941 | . 2 ⊢ Ⅎ𝑥𝐴 | |
2 | nfcv 2941 | . 2 ⊢ Ⅎ𝑥𝐵 | |
3 | 1, 2 | reueq1f 3319 | 1 ⊢ (𝐴 = 𝐵 → (∃!𝑥 ∈ 𝐴 𝜑 ↔ ∃!𝑥 ∈ 𝐵 𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1653 ∃!wreu 3091 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-ext 2777 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-tru 1657 df-ex 1876 df-nf 1880 df-mo 2591 df-eu 2609 df-cleq 2792 df-clel 2795 df-nfc 2930 df-reu 3096 |
This theorem is referenced by: reueqd 3331 lubfval 17293 glbfval 17306 uspgredg2vlem 26456 uspgredg2v 26457 isfrgr 27607 frgr1v 27620 nfrgr2v 27621 frgr3v 27624 1vwmgr 27625 3vfriswmgr 27627 isplig 27856 hdmap14lem4a 37892 hdmap14lem15 37903 |
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