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Mirrors > Home > MPE Home > Th. List > r19.21 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of 19.21 2201. (Contributed by Scott Fenton, 30-Mar-2011.) |
Ref | Expression |
---|---|
r19.21.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
r19.21 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.21.1 | . 2 ⊢ Ⅎ𝑥𝜑 | |
2 | r19.21t 3239 | . 2 ⊢ (Ⅎ𝑥𝜑 → (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓))) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 Ⅎwnf 1786 ∀wral 3065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-12 2172 |
This theorem depends on definitions: df-bi 206 df-ex 1783 df-nf 1787 df-ral 3066 |
This theorem is referenced by: rmo3f 3697 ra4 3847 rmoanim 3855 rmoanimALT 3856 r19.32 45404 |
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