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Mirrors > Home > NFE Home > Th. List > rexlimd | GIF version |
Description: Deduction from Theorem 19.23 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 27-May-1998.) (Proof shortened by Andrew Salmon, 30-May-2011.) |
Ref | Expression |
---|---|
rexlimd.1 | ⊢ Ⅎxφ |
rexlimd.2 | ⊢ Ⅎxχ |
rexlimd.3 | ⊢ (φ → (x ∈ A → (ψ → χ))) |
Ref | Expression |
---|---|
rexlimd | ⊢ (φ → (∃x ∈ A ψ → χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexlimd.1 | . . 3 ⊢ Ⅎxφ | |
2 | rexlimd.3 | . . 3 ⊢ (φ → (x ∈ A → (ψ → χ))) | |
3 | 1, 2 | ralrimi 2695 | . 2 ⊢ (φ → ∀x ∈ A (ψ → χ)) |
4 | rexlimd.2 | . . 3 ⊢ Ⅎxχ | |
5 | 4 | r19.23 2729 | . 2 ⊢ (∀x ∈ A (ψ → χ) ↔ (∃x ∈ A ψ → χ)) |
6 | 3, 5 | sylib 188 | 1 ⊢ (φ → (∃x ∈ A ψ → χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1544 ∈ wcel 1710 ∀wral 2614 ∃wrex 2615 |
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-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ex 1542 df-nf 1545 df-ral 2619 df-rex 2620 |
This theorem is referenced by: rexlimdv 2737 fun11iun 5305 ffnfv 5427 |
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