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| Mirrors > Home > MPE Home > Th. List > ceqsex2v | Structured version Visualization version GIF version | ||
| Description: Elimination of two existential quantifiers, using implicit substitution. (Contributed by Scott Fenton, 7-Jun-2006.) Avoid ax-10 2141 and ax-11 2157. (Revised by GG, 20-Aug-2023.) |
| Ref | Expression |
|---|---|
| ceqsex2v.1 | ⊢ 𝐴 ∈ V |
| ceqsex2v.2 | ⊢ 𝐵 ∈ V |
| ceqsex2v.3 | ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) |
| ceqsex2v.4 | ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) |
| Ref | Expression |
|---|---|
| ceqsex2v | ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | 3anass 1095 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) | |
| 2 | 1 | exbii 1848 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) |
| 3 | 19.42v 1953 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))) | |
| 4 | 2, 3 | bitri 275 | . . 3 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))) |
| 5 | 4 | exbii 1848 | . 2 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑))) |
| 6 | ceqsex2v.1 | . . 3 ⊢ 𝐴 ∈ V | |
| 7 | ceqsex2v.3 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓)) | |
| 8 | 7 | anbi2d 630 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑦 = 𝐵 ∧ 𝜓))) |
| 9 | 8 | exbidv 1921 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓))) |
| 10 | 6, 9 | ceqsexv 3532 | . 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓)) |
| 11 | ceqsex2v.2 | . . 3 ⊢ 𝐵 ∈ V | |
| 12 | ceqsex2v.4 | . . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒)) | |
| 13 | 11, 12 | ceqsexv 3532 | . 2 ⊢ (∃𝑦(𝑦 = 𝐵 ∧ 𝜓) ↔ 𝜒) |
| 14 | 5, 10, 13 | 3bitri 297 | 1 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ 𝜒) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 |
| 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-3an 1089 df-ex 1780 df-clel 2816 |
| This theorem is referenced by: ceqsex3v 3537 ceqsex4v 3538 ispos 18360 elfuns 35916 brimg 35938 brapply 35939 brsuccf 35942 brrestrict 35950 dfrdg4 35952 diblsmopel 41173 |
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