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Mirrors > Home > MPE Home > Th. List > ralimi2 | Structured version Visualization version GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 22-Feb-2004.) |
Ref | Expression |
---|---|
ralimi2.1 | ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓)) |
Ref | Expression |
---|---|
ralimi2 | ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimi2.1 | . . 3 ⊢ ((𝑥 ∈ 𝐴 → 𝜑) → (𝑥 ∈ 𝐵 → 𝜓)) | |
2 | 1 | alimi 1803 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) |
3 | df-ral 3140 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | df-ral 3140 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
5 | 2, 3, 4 | 3imtr4i 293 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1526 ∈ wcel 2105 ∀wral 3135 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 |
This theorem depends on definitions: df-bi 208 df-ral 3140 |
This theorem is referenced by: ralimia 3155 ralcom3 3362 tfi 7557 resixpfo 8488 omex 9094 kmlem1 9564 brdom5 9939 brdom4 9940 xrub 12693 pcmptcl 16215 itgeq2 24305 iblcnlem 24316 pntrsumbnd 26069 nmounbseqi 28481 nmounbseqiALT 28482 sumdmdi 30124 dmdbr4ati 30125 dmdbr6ati 30127 bnj110 32029 fiinfi 39810 |
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