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Mirrors > Home > NFE Home > Th. List > mprg | GIF version |
Description: Modus ponens combined with restricted generalization. (Contributed by NM, 10-Aug-2004.) |
Ref | Expression |
---|---|
mprg.1 | ⊢ (∀x ∈ A φ → ψ) |
mprg.2 | ⊢ (x ∈ A → φ) |
Ref | Expression |
---|---|
mprg | ⊢ ψ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mprg.2 | . . 3 ⊢ (x ∈ A → φ) | |
2 | 1 | rgen 2679 | . 2 ⊢ ∀x ∈ A φ |
3 | mprg.1 | . 2 ⊢ (∀x ∈ A φ → ψ) | |
4 | 2, 3 | ax-mp 8 | 1 ⊢ ψ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1710 ∀wral 2614 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 |
This theorem depends on definitions: df-bi 177 df-ral 2619 |
This theorem is referenced by: reximia 2719 rmoimia 3036 iuneq2i 3987 iineq2i 3988 dfiun2 4001 dfiin2 4002 fnpw1fn 5853 dfnnc3 5885 enmap2lem2 6064 enmap1lem2 6070 enprmaplem2 6077 enprmaplem5 6080 fntcfn 6245 fnspac 6283 fnfreclem3 6319 |
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