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Mirrors > Home > ILE 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 2368 | . 2 ⊢ ∀x ∈ A φ |
3 | mprg.1 | . 2 ⊢ (∀x ∈ A φ → ψ) | |
4 | 2, 3 | ax-mp 7 | 1 ⊢ ψ |
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-gen 1335 |
This theorem depends on definitions: df-bi 110 df-ral 2305 |
This theorem is referenced by: reximia 2408 rmoimia 2735 iuneq2i 3666 iineq2i 3667 dfiun2 3682 dfiin2 3683 dfiun3 4534 dfiin3 4535 cnviinm 4802 bj-omtrans 9416 |
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