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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 2136 and ax-12 2170. (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 3483 | 1 ⊢ (𝐴 ∈ 𝑉 → (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∃wex 1780 ∈ wcel 2105 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-clel 2814 |
This theorem is referenced by: ceqsrexv 3594 clel3g 3601 elxp5 7838 xpsnen 8920 isssc 17629 metuel2 23827 isgrpo 29147 bj-finsumval0 35561 ismgmOLD 36113 brxrn 36641 pmapjat1 38121 dfatdmfcoafv2 45097 |
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