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Mirrors > Home > MPE Home > Th. List > hbxfrbi | Structured version Visualization version GIF version |
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2865 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
hbxfrbi.1 | ⊢ (𝜑 ↔ 𝜓) |
hbxfrbi.2 | ⊢ (𝜓 → ∀𝑥𝜓) |
Ref | Expression |
---|---|
hbxfrbi | ⊢ (𝜑 → ∀𝑥𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbxfrbi.2 | . 2 ⊢ (𝜓 → ∀𝑥𝜓) | |
2 | hbxfrbi.1 | . 2 ⊢ (𝜑 ↔ 𝜓) | |
3 | 2 | albii 1822 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
4 | 1, 2, 3 | 3imtr4i 292 | 1 ⊢ (𝜑 → ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∀wal 1540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 |
This theorem depends on definitions: df-bi 206 |
This theorem is referenced by: hbn1fw 2049 hbe1w 2052 hbe1 2140 hbsbw 2170 hbab1OLD 2720 hbab 2721 hbabg 2722 hbxfreq 2865 hbral 3306 bnj982 33789 bnj1095 33792 bnj1096 33793 bnj1276 33825 bnj594 33923 bnj1445 34055 hbra2VD 43621 |
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