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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 2139 and ax-11 2155. (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 1094 | . . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ (𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) | |
2 | 1 | exbii 1845 | . . . 4 ⊢ (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵 ∧ 𝜑))) |
3 | 19.42v 1951 | . . . 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 1845 | . 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 1919 | . . 3 ⊢ (𝑥 = 𝐴 → (∃𝑦(𝑦 = 𝐵 ∧ 𝜑) ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝜓))) |
10 | 6, 9 | ceqsexv 3530 | . 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 3530 | . 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 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 |
This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-ex 1777 df-clel 2814 |
This theorem is referenced by: ceqsex3v 3537 ceqsex4v 3538 ispos 18372 elfuns 35897 brimg 35919 brapply 35920 brsuccf 35923 brrestrict 35931 dfrdg4 35933 diblsmopel 41154 |
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