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Mirrors > Home > ILE Home > Th. List > hbxfrbi | GIF version |
Description: A utility lemma to transfer a bound-variable hypothesis builder into a definition. (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 1447 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
4 | 1, 2, 3 | 3imtr4i 200 | 1 ⊢ (𝜑 → ∀𝑥𝜑) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1330 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1424 ax-gen 1426 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: hbbi 1528 hb3or 1529 hb3an 1530 hbs1f 1755 hbs1 1912 hbsbv 1915 hbeu1 2010 sb8euh 2023 hbmo1 2038 hbmo 2039 hbab1 2129 hbab 2131 cleqh 2240 hbxfreq 2247 hbral 2467 hbra1 2468 |
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