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Mirrors > Home > ILE Home > Th. List > cbvraldva2 | GIF version |
Description: Rule used to change the bound variable in a restricted universal quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
cbvraldva2.1 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) |
cbvraldva2.2 | ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
cbvraldva2 | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦) | |
2 | cbvraldva2.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
3 | 1, 2 | eleq12d 2170 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | cbvraldva2.1 | . . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | imbi12d 233 | . . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒))) |
6 | 5 | cbvaldva 1863 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒))) |
7 | df-ral 2380 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
8 | df-ral 2380 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)) | |
9 | 6, 7, 8 | 3bitr4g 222 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1297 = wceq 1299 ∈ wcel 1448 ∀wral 2375 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-cleq 2093 df-clel 2096 df-ral 2380 |
This theorem is referenced by: cbvraldva 2618 acexmid 5705 tfrlem3ag 6136 tfrlem3a 6137 tfrlemi1 6159 tfr1onlem3ag 6164 |
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