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Mirrors > Home > NFE Home > Th. List > mprgbir | GIF version |
Description: Modus ponens on biconditional combined with restricted generalization. (Contributed by NM, 21-Mar-2004.) |
Ref | Expression |
---|---|
mprgbir.1 | ⊢ (φ ↔ ∀x ∈ A ψ) |
mprgbir.2 | ⊢ (x ∈ A → ψ) |
Ref | Expression |
---|---|
mprgbir | ⊢ φ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mprgbir.2 | . . 3 ⊢ (x ∈ A → ψ) | |
2 | 1 | rgen 2679 | . 2 ⊢ ∀x ∈ A ψ |
3 | mprgbir.1 | . 2 ⊢ (φ ↔ ∀x ∈ A ψ) | |
4 | 2, 3 | mpbir 200 | 1 ⊢ φ |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∈ 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: ss2rabi 3348 rabxm 3573 rabnc 3574 ssintub 3944 pw10 4161 opeq 4619 dmiin 4965 xpnedisj 5513 |
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