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Mirrors > Home > MPE Home > Th. List > rspec2 | Structured version Visualization version GIF version |
Description: Specialization rule for restricted quantification, with two quantifiers. (Contributed by NM, 20-Nov-1994.) |
Ref | Expression |
---|---|
rspec2.1 | ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 |
Ref | Expression |
---|---|
rspec2 | ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rspec2.1 | . . 3 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝜑 | |
2 | 1 | rspec 3209 | . 2 ⊢ (𝑥 ∈ 𝐴 → ∀𝑦 ∈ 𝐵 𝜑) |
3 | 2 | r19.21bi 3210 | 1 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-12 2177 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-ral 3145 |
This theorem is referenced by: rspec3 3214 |
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