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Mirrors > Home > MPE Home > Th. List > mpgbi | Structured version Visualization version 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 1870 | . 2 ⊢ ∀𝑥𝜑 |
3 | mpgbi.1 | . 2 ⊢ (∀𝑥𝜑 ↔ 𝜓) | |
4 | 2, 3 | mpbi 220 | 1 ⊢ 𝜓 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∀wal 1629 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 |
This theorem depends on definitions: df-bi 197 |
This theorem is referenced by: nex 1879 exlimi 2242 exlimiOLD 2383 axi12 2749 abbii 2888 nalset 4929 bnj1304 31228 bnj1052 31381 bnj1030 31393 bj-abbii 33113 bj-nalset 33130 bj-nuliota 33350 spr0nelg 42254 |
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