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Mirrors > Home > ILE Home > Th. List > ceqsrexbv | GIF version |
Description: Elimination of a restricted existential quantifier, using implicit substitution. (Contributed by Mario Carneiro, 14-Mar-2014.) |
Ref | Expression |
---|---|
ceqsrexv.1 | ⊢ (x = A → (φ ↔ ψ)) |
Ref | Expression |
---|---|
ceqsrexbv | ⊢ (∃x ∈ B (x = A ∧ φ) ↔ (A ∈ B ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.42v 2461 | . 2 ⊢ (∃x ∈ B (A ∈ B ∧ (x = A ∧ φ)) ↔ (A ∈ B ∧ ∃x ∈ B (x = A ∧ φ))) | |
2 | eleq1 2097 | . . . . . . 7 ⊢ (x = A → (x ∈ B ↔ A ∈ B)) | |
3 | 2 | adantr 261 | . . . . . 6 ⊢ ((x = A ∧ φ) → (x ∈ B ↔ A ∈ B)) |
4 | 3 | pm5.32ri 428 | . . . . 5 ⊢ ((x ∈ B ∧ (x = A ∧ φ)) ↔ (A ∈ B ∧ (x = A ∧ φ))) |
5 | 4 | bicomi 123 | . . . 4 ⊢ ((A ∈ B ∧ (x = A ∧ φ)) ↔ (x ∈ B ∧ (x = A ∧ φ))) |
6 | 5 | baib 827 | . . 3 ⊢ (x ∈ B → ((A ∈ B ∧ (x = A ∧ φ)) ↔ (x = A ∧ φ))) |
7 | 6 | rexbiia 2333 | . 2 ⊢ (∃x ∈ B (A ∈ B ∧ (x = A ∧ φ)) ↔ ∃x ∈ B (x = A ∧ φ)) |
8 | ceqsrexv.1 | . . . 4 ⊢ (x = A → (φ ↔ ψ)) | |
9 | 8 | ceqsrexv 2668 | . . 3 ⊢ (A ∈ B → (∃x ∈ B (x = A ∧ φ) ↔ ψ)) |
10 | 9 | pm5.32i 427 | . 2 ⊢ ((A ∈ B ∧ ∃x ∈ B (x = A ∧ φ)) ↔ (A ∈ B ∧ ψ)) |
11 | 1, 7, 10 | 3bitr3i 199 | 1 ⊢ (∃x ∈ B (x = A ∧ φ) ↔ (A ∈ B ∧ ψ)) |
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: frecsuclem3 5929 |
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