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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 2759 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 1787 | . 2 ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) |
4 | 1, 2, 3 | 3imtr4i 281 | 1 ⊢ (𝜑 → ∀𝑥𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∀wal 1521 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 |
This theorem depends on definitions: df-bi 197 |
This theorem is referenced by: hbn1fw 2014 hbe1w 2018 hbe1 2061 hbexOLD 2190 hbab1 2640 hbab 2642 hbxfreq 2759 hbral 2972 bnj982 30975 bnj1095 30978 bnj1096 30979 bnj1276 31011 bnj594 31108 bnj1445 31238 bj-hbab1 32896 hbra2VD 39410 |
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