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Mirrors > Home > ILE Home > Th. List > albidh | GIF version |
Description: Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
albidh.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
albidh.2 | ⊢ (𝜑 → (𝜓 ↔ 𝜒)) |
Ref | Expression |
---|---|
albidh | ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | albidh.1 | . . 3 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | albidh.2 | . . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒)) | |
3 | 1, 2 | alrimih 1469 | . 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒)) |
4 | albi 1468 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | |
5 | 3, 4 | syl 14 | 1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-gen 1449 |
This theorem depends on definitions: df-bi 117 |
This theorem is referenced by: nfbidf 1539 albid 1615 dral2 1731 ax11v2 1820 albidv 1824 equs5or 1830 sbal2 2020 eubidh 2032 |
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