Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 1730 | . 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 → 𝜑) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) |
3 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜑)) | |
4 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓)) | |
5 | 2, 3, 4 | 3imtr4i 280 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝜑 → ∀𝑥 ∈ 𝐵 𝜓) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1473 ∈ wcel 1977 ∀wral 2896 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 |
This theorem depends on definitions: df-bi 196 df-ral 2901 |
This theorem is referenced by: ralimia 2934 ralcom3 3084 tfi 6945 resixpfo 7832 omex 8423 kmlem1 8855 brdom5 9232 brdom4 9233 xrub 12014 pcmptcl 15433 itgeq2 23350 iblcnlem 23361 pntrsumbnd 25055 nmounbseqi 27016 nmounbseqiALT 27017 sumdmdi 28663 dmdbr4ati 28664 dmdbr6ati 28666 bnj110 30182 fiinfi 36897 |
Copyright terms: Public domain | W3C validator |