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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 261 | . 2 ⊢ ((φ ∧ x ∈ A) → (ψ → χ)) |
3 | 2 | ralimdva 2381 | 1 ⊢ (φ → (∀x ∈ A ψ → ∀x ∈ A χ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1390 ∀wral 2300 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-4 1397 ax-17 1416 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-ral 2305 |
This theorem is referenced by: poss 4026 sess1 4059 sess2 4060 riinint 4536 dffo4 5258 dffo5 5259 isoini2 5401 rdgivallem 5908 iinerm 6114 |
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