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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 2173. (Revised by Gino Giotto, 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 3537 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1777 df-cleq 2814 df-clel 2893 |
This theorem is referenced by: ceqsrexv 3648 clel3g 3653 elxp5 7622 xpsnen 8595 isssc 17084 metuel2 23169 isgrpo 28268 bj-finsumval0 34561 ismgmOLD 35122 brxrn 35620 pmapjat1 36983 dfatdmfcoafv2 43447 |
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