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Mirrors > Home > MPE Home > Th. List > ceqsexv2d | Structured version Visualization version GIF version |
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.) |
Ref | Expression |
---|---|
ceqsexv2d.1 | ⊢ 𝐴 ∈ V |
ceqsexv2d.2 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
ceqsexv2d.3 | ⊢ 𝜓 |
Ref | Expression |
---|---|
ceqsexv2d | ⊢ ∃𝑥𝜑 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ceqsexv2d.3 | . 2 ⊢ 𝜓 | |
2 | ceqsexv2d.1 | . . . 4 ⊢ 𝐴 ∈ V | |
3 | ceqsexv2d.2 | . . . 4 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
4 | 2, 3 | ceqsexv 3215 | . . 3 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) ↔ 𝜓) |
5 | 4 | biimpri 217 | . 2 ⊢ (𝜓 → ∃𝑥(𝑥 = 𝐴 ∧ 𝜑)) |
6 | exsimpr 1784 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝜑) → ∃𝑥𝜑) | |
7 | 1, 5, 6 | mp2b 10 | 1 ⊢ ∃𝑥𝜑 |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-12 2034 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-v 3175 |
This theorem is referenced by: 2lgslem1 24919 griedg0prc 40488 1loopgrvd2 40718 |
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