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Mirrors > Home > ILE Home > Th. List > cbvrexdva2 | 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.) |
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 103 | . . . . 5 ⊢ ((φ ∧ x = y) → x = y) | |
2 | cbvraldva2.2 | . . . . 5 ⊢ ((φ ∧ x = y) → A = B) | |
3 | 1, 2 | eleq12d 2105 | . . . 4 ⊢ ((φ ∧ x = y) → (x ∈ A ↔ y ∈ B)) |
4 | cbvraldva2.1 | . . . 4 ⊢ ((φ ∧ x = y) → (ψ ↔ χ)) | |
5 | 3, 4 | anbi12d 442 | . . 3 ⊢ ((φ ∧ x = y) → ((x ∈ A ∧ ψ) ↔ (y ∈ B ∧ χ))) |
6 | 5 | cbvexdva 1801 | . 2 ⊢ (φ → (∃x(x ∈ A ∧ ψ) ↔ ∃y(y ∈ B ∧ χ))) |
7 | df-rex 2306 | . 2 ⊢ (∃x ∈ A ψ ↔ ∃x(x ∈ A ∧ ψ)) | |
8 | df-rex 2306 | . 2 ⊢ (∃y ∈ B χ ↔ ∃y(y ∈ B ∧ χ)) | |
9 | 6, 7, 8 | 3bitr4g 212 | 1 ⊢ (φ → (∃x ∈ A ψ ↔ ∃y ∈ B χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-cleq 2030 df-clel 2033 df-rex 2306 |
This theorem is referenced by: cbvrexdva 2534 acexmid 5454 |
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