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Mirrors > Home > ILE Home > Th. List > mpgbi | GIF version |
Description: Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.) |
Ref | Expression |
---|---|
mpgbi.1 | ⊢ (∀𝑥𝜑 ↔ 𝜓) |
mpgbi.2 | ⊢ 𝜑 |
Ref | Expression |
---|---|
mpgbi | ⊢ 𝜓 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpgbi.2 | . . 3 ⊢ 𝜑 | |
2 | 1 | ax-gen 1338 | . 2 ⊢ ∀𝑥𝜑 |
3 | mpgbi.1 | . 2 ⊢ (∀𝑥𝜑 ↔ 𝜓) | |
4 | 2, 3 | mpbi 133 | 1 ⊢ 𝜓 |
Colors of variables: wff set class |
Syntax hints: ↔ wb 98 ∀wal 1241 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-gen 1338 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: nex 1389 exlimih 1484 exan 1583 abbii 2153 bj-ex 9902 |
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