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Mirrors > Home > MPE Home > Th. List > cbvraldva2 | Structured version Visualization version 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 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝑥 = 𝑦) | |
2 | cbvraldva2.2 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → 𝐴 = 𝐵) | |
3 | 1, 2 | eleq12d 2907 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐵)) |
4 | cbvraldva2.1 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒)) | |
5 | 3, 4 | imbi12d 347 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒))) |
6 | 5 | expcom 416 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) ↔ (𝑦 ∈ 𝐵 → 𝜒)))) |
7 | 6 | pm5.74d 275 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → (𝑦 ∈ 𝐵 → 𝜒)))) |
8 | 7 | cbvalvw 2043 | . . . 4 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ ∀𝑦(𝜑 → (𝑦 ∈ 𝐵 → 𝜒))) |
9 | 19.21v 1940 | . . . 4 ⊢ (∀𝑥(𝜑 → (𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓))) | |
10 | 19.21v 1940 | . . . 4 ⊢ (∀𝑦(𝜑 → (𝑦 ∈ 𝐵 → 𝜒)) ↔ (𝜑 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜒))) | |
11 | 8, 9, 10 | 3bitr3i 303 | . . 3 ⊢ ((𝜑 → ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) ↔ (𝜑 → ∀𝑦(𝑦 ∈ 𝐵 → 𝜒))) |
12 | 11 | pm5.74ri 274 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒))) |
13 | df-ral 3143 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
14 | df-ral 3143 | . 2 ⊢ (∀𝑦 ∈ 𝐵 𝜒 ↔ ∀𝑦(𝑦 ∈ 𝐵 → 𝜒)) | |
15 | 12, 13, 14 | 3bitr4g 316 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑦 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∀wal 1535 = wceq 1537 ∈ wcel 2114 ∀wral 3138 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2814 df-clel 2893 df-ral 3143 |
This theorem is referenced by: cbvraldva 3459 tfrlem3a 8013 mreexexlemd 16915 ismnu 40617 |
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