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Mirrors > Home > ILE Home > Th. List > ralimdv | GIF version |
Description: Deduction quantifying both antecedent and consequent, based on Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 8-Oct-2003.) |
Ref | Expression |
---|---|
ralimdv.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ralimdv | ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralimdv.1 | . . 3 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
2 | 1 | adantr 274 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝜓 → 𝜒)) |
3 | 2 | ralimdva 2502 | 1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐴 𝜓 → ∀𝑥 ∈ 𝐴 𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ∀wral 2417 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 ax-4 1488 ax-17 1507 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-ral 2422 |
This theorem is referenced by: poss 4228 sess1 4267 sess2 4268 riinint 4808 dffo4 5576 dffo5 5577 isoini2 5728 rdgivallem 6286 iinerm 6509 xpf1o 6746 resqrexlemgt0 10824 cau3lem 10918 caubnd2 10921 climshftlemg 11103 climcau 11148 climcaucn 11152 serf0 11153 modfsummodlemstep 11258 bezoutlemmain 11722 ctinf 11979 strsetsid 12031 fiinbas 12255 baspartn 12256 lmtopcnp 12458 rescncf 12776 limcresi 12843 |
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