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Mirrors > Home > NFE Home > Th. List > elrab2 | GIF version |
Description: Membership in a class abstraction, using implicit substitution. (Contributed by NM, 2-Nov-2006.) |
Ref | Expression |
---|---|
elrab2.1 | ⊢ (x = A → (φ ↔ ψ)) |
elrab2.2 | ⊢ C = {x ∈ B ∣ φ} |
Ref | Expression |
---|---|
elrab2 | ⊢ (A ∈ C ↔ (A ∈ B ∧ ψ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elrab2.2 | . . 3 ⊢ C = {x ∈ B ∣ φ} | |
2 | 1 | eleq2i 2417 | . 2 ⊢ (A ∈ C ↔ A ∈ {x ∈ B ∣ φ}) |
3 | elrab2.1 | . . 3 ⊢ (x = A → (φ ↔ ψ)) | |
4 | 3 | elrab 2994 | . 2 ⊢ (A ∈ {x ∈ B ∣ φ} ↔ (A ∈ B ∧ ψ)) |
5 | 2, 4 | bitri 240 | 1 ⊢ (A ∈ C ↔ (A ∈ B ∧ ψ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 = wceq 1642 ∈ wcel 1710 {crab 2618 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-rab 2623 df-v 2861 |
This theorem is referenced by: elrabsf 3084 fvmpti 5699 fvmptss2 5725 nenpw1pwlem2 6085 |
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