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Mirrors > Home > ILE Home > Th. List > albi | GIF version |
Description: Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
albi | ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bi1 117 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | |
2 | 1 | al2imi 1419 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) |
3 | bi2 129 | . . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | |
4 | 3 | al2imi 1419 | . 2 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜓 → ∀𝑥𝜑)) |
5 | 2, 4 | impbid 128 | 1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1314 |
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 1408 ax-gen 1410 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: albii 1431 albidh 1441 19.16 1519 19.17 1520 intmin4 3769 dfiin2g 3816 |
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