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Mirrors > Home > NFE Home > Th. List > rexpr | GIF version |
Description: Convert an existential quantification over a pair to a disjunction. (Contributed by NM, 3-Jun-2007.) (Revised by Mario Carneiro, 23-Apr-2015.) |
Ref | Expression |
---|---|
ralpr.1 | ⊢ A ∈ V |
ralpr.2 | ⊢ B ∈ V |
ralpr.3 | ⊢ (x = A → (φ ↔ ψ)) |
ralpr.4 | ⊢ (x = B → (φ ↔ χ)) |
Ref | Expression |
---|---|
rexpr | ⊢ (∃x ∈ {A, B}φ ↔ (ψ ∨ χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralpr.1 | . 2 ⊢ A ∈ V | |
2 | ralpr.2 | . 2 ⊢ B ∈ V | |
3 | ralpr.3 | . . 3 ⊢ (x = A → (φ ↔ ψ)) | |
4 | ralpr.4 | . . 3 ⊢ (x = B → (φ ↔ χ)) | |
5 | 3, 4 | rexprg 3776 | . 2 ⊢ ((A ∈ V ∧ B ∈ V) → (∃x ∈ {A, B}φ ↔ (ψ ∨ χ))) |
6 | 1, 2, 5 | mp2an 653 | 1 ⊢ (∃x ∈ {A, B}φ ↔ (ψ ∨ χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∨ wo 357 = wceq 1642 ∈ wcel 1710 ∃wrex 2615 Vcvv 2859 {cpr 3738 |
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-3an 936 df-nan 1288 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-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 |
This theorem is referenced by: (None) |
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