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Mirrors > Home > MPE Home > Th. List > cbvrexdva2 | Structured version Visualization version GIF version |
Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) (Proof shortened by Wolf Lammen, 12-Aug-2023.) |
Ref | Expression |
---|---|
cbvraldva2.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
cbvraldva2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvrexdva2 | ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦) | |
2 | cbvraldva2.2 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
3 | 1, 2 | eleq12d 2833 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | cbvraldva2.1 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | anbi12d 631 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒))) |
6 | 5 | ancoms 459 | . . . . . 6 ⊢ ((𝑥 = 𝑦 ∧ 𝜑) → ((𝑥 ∈ 𝐴 ∧ 𝜓) ↔ (𝑦 ∈ 𝐵 ∧ 𝜒))) |
7 | 6 | pm5.32da 579 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜒)))) |
8 | 7 | cbvexvw 2040 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ ∃𝑦(𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜒))) |
9 | 19.42v 1957 | . . . 4 ⊢ (∃𝑥(𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓))) | |
10 | 19.42v 1957 | . . . 4 ⊢ (∃𝑦(𝜑 ∧ (𝑦 ∈ 𝐵 ∧ 𝜒)) ↔ (𝜑 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))) | |
11 | 8, 9, 10 | 3bitr3i 301 | . . 3 ⊢ ((𝜑 ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))) |
12 | pm5.32 574 | . . 3 ⊢ ((𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))) ↔ ((𝜑 ∧ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) ↔ (𝜑 ∧ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)))) | |
13 | 11, 12 | mpbir 230 | . 2 ⊢ (𝜑 → (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒))) |
14 | df-rex 3070 | . 2 ⊢ (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝜓)) | |
15 | df-rex 3070 | . 2 ⊢ (∃𝑦 ∈ 𝐵 𝜒 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜒)) | |
16 | 13, 14, 15 | 3bitr4g 314 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∃wex 1782 ∈ wcel 2106 ∃wrex 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1783 df-cleq 2730 df-clel 2816 df-rex 3070 |
This theorem is referenced by: cbvrexdva 3392 mreexexlemd 17363 eulerpartlemgvv 32351 ismnu 41860 |
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