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Mirrors > Home > ILE Home > Th. List > r19.21be | 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 2407 | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝜓) |
3 | 2 | expcom 109 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
4 | 3 | rgen 2374 | 1 ⊢ ∀𝑥 ∈ 𝐴 (𝜑 → 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1393 ∀wral 2306 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-gen 1338 ax-4 1400 |
This theorem depends on definitions: df-bi 110 df-ral 2311 |
This theorem is referenced by: (None) |
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