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Mirrors > Home > ILE Home > Th. List > hbbid | GIF version |
Description: Deduction form of bound-variable hypothesis builder hbbi 1528. (Contributed by NM, 1-Jan-2002.) |
Ref | Expression |
---|---|
hbbid.1 | ⊢ (𝜑 → ∀𝑥𝜑) |
hbbid.2 | ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) |
hbbid.3 | ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) |
Ref | Expression |
---|---|
hbbid | ⊢ (𝜑 → ((𝜓 ↔ 𝜒) → ∀𝑥(𝜓 ↔ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hbbid.1 | . . . 4 ⊢ (𝜑 → ∀𝑥𝜑) | |
2 | hbbid.2 | . . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓)) | |
3 | hbbid.3 | . . . 4 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒)) | |
4 | 1, 2, 3 | hbimd 1553 | . . 3 ⊢ (𝜑 → ((𝜓 → 𝜒) → ∀𝑥(𝜓 → 𝜒))) |
5 | 1, 3, 2 | hbimd 1553 | . . 3 ⊢ (𝜑 → ((𝜒 → 𝜓) → ∀𝑥(𝜒 → 𝜓))) |
6 | 4, 5 | anim12d 333 | . 2 ⊢ (𝜑 → (((𝜓 → 𝜒) ∧ (𝜒 → 𝜓)) → (∀𝑥(𝜓 → 𝜒) ∧ ∀𝑥(𝜒 → 𝜓)))) |
7 | dfbi2 386 | . 2 ⊢ ((𝜓 ↔ 𝜒) ↔ ((𝜓 → 𝜒) ∧ (𝜒 → 𝜓))) | |
8 | albiim 1467 | . 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) ↔ (∀𝑥(𝜓 → 𝜒) ∧ ∀𝑥(𝜒 → 𝜓))) | |
9 | 6, 7, 8 | 3imtr4g 204 | 1 ⊢ (𝜑 → ((𝜓 ↔ 𝜒) → ∀𝑥(𝜓 ↔ 𝜒))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∀wal 1333 |
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 1427 ax-gen 1429 ax-4 1490 ax-i5r 1515 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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