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Mirrors > Home > ILE Home > Th. List > ceqsrex2v | GIF version |
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by NM, 29-Oct-2005.) |
Ref | Expression |
---|---|
ceqsrex2v.1 | ⊢ (x = A → (φ ↔ ψ)) |
ceqsrex2v.2 | ⊢ (y = B → (ψ ↔ χ)) |
Ref | Expression |
---|---|
ceqsrex2v | ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (∃x ∈ 𝐶 ∃y ∈ 𝐷 ((x = A ∧ y = B) ∧ φ) ↔ χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | anass 381 | . . . . . 6 ⊢ (((x = A ∧ y = B) ∧ φ) ↔ (x = A ∧ (y = B ∧ φ))) | |
2 | 1 | rexbii 2325 | . . . . 5 ⊢ (∃y ∈ 𝐷 ((x = A ∧ y = B) ∧ φ) ↔ ∃y ∈ 𝐷 (x = A ∧ (y = B ∧ φ))) |
3 | r19.42v 2461 | . . . . 5 ⊢ (∃y ∈ 𝐷 (x = A ∧ (y = B ∧ φ)) ↔ (x = A ∧ ∃y ∈ 𝐷 (y = B ∧ φ))) | |
4 | 2, 3 | bitri 173 | . . . 4 ⊢ (∃y ∈ 𝐷 ((x = A ∧ y = B) ∧ φ) ↔ (x = A ∧ ∃y ∈ 𝐷 (y = B ∧ φ))) |
5 | 4 | rexbii 2325 | . . 3 ⊢ (∃x ∈ 𝐶 ∃y ∈ 𝐷 ((x = A ∧ y = B) ∧ φ) ↔ ∃x ∈ 𝐶 (x = A ∧ ∃y ∈ 𝐷 (y = B ∧ φ))) |
6 | ceqsrex2v.1 | . . . . . 6 ⊢ (x = A → (φ ↔ ψ)) | |
7 | 6 | anbi2d 437 | . . . . 5 ⊢ (x = A → ((y = B ∧ φ) ↔ (y = B ∧ ψ))) |
8 | 7 | rexbidv 2321 | . . . 4 ⊢ (x = A → (∃y ∈ 𝐷 (y = B ∧ φ) ↔ ∃y ∈ 𝐷 (y = B ∧ ψ))) |
9 | 8 | ceqsrexv 2668 | . . 3 ⊢ (A ∈ 𝐶 → (∃x ∈ 𝐶 (x = A ∧ ∃y ∈ 𝐷 (y = B ∧ φ)) ↔ ∃y ∈ 𝐷 (y = B ∧ ψ))) |
10 | 5, 9 | syl5bb 181 | . 2 ⊢ (A ∈ 𝐶 → (∃x ∈ 𝐶 ∃y ∈ 𝐷 ((x = A ∧ y = B) ∧ φ) ↔ ∃y ∈ 𝐷 (y = B ∧ ψ))) |
11 | ceqsrex2v.2 | . . 3 ⊢ (y = B → (ψ ↔ χ)) | |
12 | 11 | ceqsrexv 2668 | . 2 ⊢ (B ∈ 𝐷 → (∃y ∈ 𝐷 (y = B ∧ ψ) ↔ χ)) |
13 | 10, 12 | sylan9bb 435 | 1 ⊢ ((A ∈ 𝐶 ∧ B ∈ 𝐷) → (∃x ∈ 𝐶 ∃y ∈ 𝐷 ((x = A ∧ y = B) ∧ φ) ↔ χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 = wceq 1242 ∈ 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-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-rex 2306 df-v 2553 |
This theorem is referenced by: (None) |
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