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Mirrors > Home > MPE Home > Th. List > r19.21be | Structured version Visualization version GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90. (Restricted quantifier version.) (Contributed by NM, 21-Nov-1994.) |
Ref | Expression |
---|---|
r19.21be.1 | ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Ref | Expression |
---|---|
r19.21be | ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | r19.21be.1 | . . . 4 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝜓) | |
2 | 1 | r19.21bi 3135 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
3 | 2 | expcom 414 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
4 | 3 | rgen 3076 | 1 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2110 ∀wral 3066 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-12 2175 |
This theorem depends on definitions: df-bi 206 df-an 397 df-ex 1787 df-ral 3071 |
This theorem is referenced by: bnj580 32887 |
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