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Mirrors > Home > ILE Home > Th. List > ralimia | GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 19-Jul-1996.) |
Ref | Expression |
---|---|
ralimia.1 | ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) |
Ref | Expression |
---|---|
ralimia | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimia.1 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (𝜑 → 𝜓)) | |
2 | 1 | a2i 11 | . 2 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐴 → 𝜓)) |
3 | 2 | ralimi2 2537 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐴 𝜓) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2148 ∀wral 2455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 |
This theorem depends on definitions: df-bi 117 df-ral 2460 |
This theorem is referenced by: ralimiaa 2539 ralimi 2540 r19.12 2583 rr19.3v 2878 rr19.28v 2879 ffvresb 5681 f1mpt 5774 ixpf 6722 exmidontri2or 7244 peano2nnnn 7854 peano5nnnn 7893 peano5nni 8924 peano2nn 8933 serf0 11362 baspartn 13589 tridceq 14843 |
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