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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 1832 | . 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐴 → 𝜓) → ∀𝑥(𝑥 ∈ 𝐵 → 𝜒))) |
3 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝜓)) | |
4 | df-ral 2901 | . 2 ⊢ (∀𝑥 ∈ 𝐵 𝜒 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜒)) | |
5 | 2, 3, 4 | 3imtr4g 284 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐵 𝜒)) |
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 ax-5 1827 |
This theorem depends on definitions: df-bi 196 df-ral 2901 |
This theorem is referenced by: ralimdva 2945 ssralv 3629 zorn2lem7 9207 pwfseqlem3 9361 sup2 10858 xrsupexmnf 12007 xrinfmexpnf 12008 xrsupsslem 12009 xrinfmsslem 12010 xrub 12014 r19.29uz 13938 rexuzre 13940 caurcvg 14255 caucvg 14257 isprm5 15257 prmgaplem5 15597 prmgaplem6 15598 mrissmrid 16124 elcls3 20697 iscnp4 20877 cncls2 20887 cnntr 20889 2ndcsep 21072 dyadmbllem 23173 xrlimcnp 24495 pntlem3 25098 sigaclfu2 29511 mapdordlem2 35944 iunrelexp0 37013 climrec 38670 0ellimcdiv 38716 |
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