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Mirrors > Home > MPE Home > Th. List > ra4 | Structured version Visualization version GIF version |
Description: Restricted quantifier version of Axiom 5 of [Mendelson] p. 69. This is the axiom stdpc5 2201 of standard predicate calculus for a restricted domain. See ra4v 3866 for a version requiring fewer axioms. (Contributed by NM, 16-Jan-2004.) (Proof shortened by BJ, 27-Mar-2020.) |
Ref | Expression |
---|---|
ra4.1 | ⊢ Ⅎ𝑥𝜑 |
Ref | Expression |
---|---|
ra4 | ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ra4.1 | . . 3 ⊢ Ⅎ𝑥𝜑 | |
2 | 1 | r19.21 3213 | . 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) ↔ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
3 | 2 | biimpi 218 | 1 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) → (𝜑 → ∀𝑥 ∈ 𝐴 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 Ⅎwnf 1778 ∀wral 3136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-12 2170 |
This theorem depends on definitions: df-bi 209 df-ex 1775 df-nf 1779 df-ral 3141 |
This theorem is referenced by: (None) |
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