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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, 8-Jan-2025.) |
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 | cbvraldva2.1 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
2 | 1 | notbid 318 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (¬ 𝜓 ↔ ¬ 𝜒)) |
3 | cbvraldva2.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
4 | 2, 3 | cbvraldva2 3345 | . . 3 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝜒)) |
5 | ralnex 3073 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜓 ↔ ¬ ∃𝑥 ∈ 𝐴 𝜓) | |
6 | ralnex 3073 | . . 3 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝜒 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜒) | |
7 | 4, 5, 6 | 3bitr3g 313 | . 2 ⊢ (𝜑 → (¬ ∃𝑥 ∈ 𝐴 𝜓 ↔ ¬ ∃𝑦 ∈ 𝐵 𝜒)) |
8 | 7 | con4bid 317 | 1 ⊢ (𝜑 → (∃𝑥 ∈ 𝐴 𝜓 ↔ ∃𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∀wral 3062 ∃wrex 3071 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-ex 1783 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 |
This theorem is referenced by: cbvrexdvaOLD 3349 mreexexlemd 17588 eulerpartlemgvv 33375 ismnu 43020 |
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