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| Mirrors > Home > MPE Home > Th. List > ceqsexgv | Structured version Visualization version GIF version | ||
| Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 29-Dec-1996.) Drop ax-10 2141 and ax-12 2177. (Revised by GG, 1-Dec-2023.) |
| Ref | Expression |
|---|---|
| ceqsexgv.1 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| Ref | Expression |
|---|---|
| ceqsexgv | ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | id 22 | . 2 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴) | |
| 2 | ceqsexgv.1 | . 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 3 | 1, 2 | cgsexg 3526 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-clel 2816 |
| This theorem is referenced by: ceqsrexv 3655 clel3g 3661 elxp5 7945 xpsnen 9095 isssc 17864 metuel2 24578 isgrpo 30516 bj-finsumval0 37286 ismgmOLD 37857 brxrn 38375 pmapjat1 39855 dfatdmfcoafv2 47266 |
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