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Mirrors > Home > NFE Home > Th. List > ralrimi | GIF version |
Description: Inference from Theorem 19.21 of [Margaris] p. 90 (restricted quantifier version). (Contributed by NM, 10-Oct-1999.) |
Ref | Expression |
---|---|
ralrimi.1 | ⊢ Ⅎxφ |
ralrimi.2 | ⊢ (φ → (x ∈ A → ψ)) |
Ref | Expression |
---|---|
ralrimi | ⊢ (φ → ∀x ∈ A ψ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ralrimi.1 | . . 3 ⊢ Ⅎxφ | |
2 | ralrimi.2 | . . 3 ⊢ (φ → (x ∈ A → ψ)) | |
3 | 1, 2 | alrimi 1765 | . 2 ⊢ (φ → ∀x(x ∈ A → ψ)) |
4 | df-ral 2619 | . 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ)) | |
5 | 3, 4 | sylibr 203 | 1 ⊢ (φ → ∀x ∈ A ψ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∀wal 1540 Ⅎwnf 1544 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-11 1746 |
This theorem depends on definitions: df-bi 177 df-ex 1542 df-nf 1545 df-ral 2619 |
This theorem is referenced by: ralrimiv 2696 reximdai 2722 r19.12 2727 rexlimd 2735 rexlimd2 2736 r19.37 2760 ralcom2 2775 2rmorex 3040 iineq2d 3989 ncfinraise 4481 mpteq2da 5666 |
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