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Mirrors > Home > MPE Home > Th. List > ralimdv2 | Structured version Visualization version GIF version |
Description: Inference quantifying both antecedent and consequent. (Contributed by NM, 1-Feb-2005.) |
Ref | Expression |
---|---|
ralimdv2.1 | ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐵 → 𝜒))) |
Ref | Expression |
---|---|
ralimdv2 | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimdv2.1 | . . 3 ⊢ (𝜑 → ((𝑥 ∈ 𝐴 → 𝜓) → (𝑥 ∈ 𝐵 → 𝜒))) | |
2 | 1 | alimdv 1845 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) |
3 | df-ral 2917 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | df-ral 2917 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)) | |
5 | 2, 3, 4 | 3imtr4g 285 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1481 ∈ wcel 1990 ∀wral 2912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 |
This theorem depends on definitions: df-bi 197 df-ral 2917 |
This theorem is referenced by: ralimdva 2962 ssralv 3666 zorn2lem7 9324 pwfseqlem3 9482 sup2 10979 xrsupexmnf 12135 xrinfmexpnf 12136 xrsupsslem 12137 xrinfmsslem 12138 xrub 12142 r19.29uz 14090 rexuzre 14092 caurcvg 14407 caucvg 14409 isprm5 15419 prmgaplem5 15759 prmgaplem6 15760 mrissmrid 16301 elcls3 20887 iscnp4 21067 cncls2 21077 cnntr 21079 2ndcsep 21262 dyadmbllem 23367 xrlimcnp 24695 pntlem3 25298 sigaclfu2 30184 mapdordlem2 36926 iunrelexp0 37994 climrec 39835 0ellimcdiv 39881 |
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