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Mirrors > Home > NFE Home > Th. List > rspec3 | GIF version |
Description: Specialization rule for restricted quantification. (Contributed by NM, 20-Nov-1994.) |
Ref | Expression |
---|---|
rspec3.1 | ⊢ ∀x ∈ A ∀y ∈ B ∀z ∈ C φ |
Ref | Expression |
---|---|
rspec3 | ⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspec3.1 | . . . 4 ⊢ ∀x ∈ A ∀y ∈ B ∀z ∈ C φ | |
2 | 1 | rspec2 2713 | . . 3 ⊢ ((x ∈ A ∧ y ∈ B) → ∀z ∈ C φ) |
3 | 2 | r19.21bi 2712 | . 2 ⊢ (((x ∈ A ∧ y ∈ B) ∧ z ∈ C) → φ) |
4 | 3 | 3impa 1146 | 1 ⊢ ((x ∈ A ∧ y ∈ B ∧ z ∈ C) → φ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 ∧ w3a 934 ∈ wcel 1710 ∀wral 2614 |
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-11 1746 |
This theorem depends on definitions: df-bi 177 df-an 360 df-3an 936 df-ex 1542 df-ral 2619 |
This theorem is referenced by: (None) |
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