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| 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 1794 | . 2 ⊢ ∀𝑥𝜑 | 
| 3 | mpgbi.1 | . 2 ⊢ (∀𝑥𝜑 ↔ 𝜓) | |
| 4 | 2, 3 | mpbi 230 | 1 ⊢ 𝜓 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∀wal 1537 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 | 
| This theorem depends on definitions: df-bi 207 | 
| This theorem is referenced by: nex 1799 exlimi 2216 axi12 2705 axbnd 2706 nalset 5312 bnj1304 34834 bnj1052 34990 bnj1030 35002 bj-nuliota 37059 eu6w 42691 spr0nelg 47468 | 
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