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Mirrors > Home > ILE Home > Th. List > hbbid | GIF version |
Description: Deduction form of bound-variable hypothesis builder hbbi 1437. (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 1462 | . . 3 ⊢ (φ → ((ψ → χ) → ∀x(ψ → χ))) |
5 | 1, 3, 2 | hbimd 1462 | . . 3 ⊢ (φ → ((χ → ψ) → ∀x(χ → ψ))) |
6 | 4, 5 | anim12d 318 | . 2 ⊢ (φ → (((ψ → χ) ∧ (χ → ψ)) → (∀x(ψ → χ) ∧ ∀x(χ → ψ)))) |
7 | dfbi2 368 | . 2 ⊢ ((ψ ↔ χ) ↔ ((ψ → χ) ∧ (χ → ψ))) | |
8 | albiim 1373 | . 2 ⊢ (∀x(ψ ↔ χ) ↔ (∀x(ψ → χ) ∧ ∀x(χ → ψ))) | |
9 | 6, 7, 8 | 3imtr4g 194 | 1 ⊢ (φ → ((ψ ↔ χ) → ∀x(ψ ↔ χ))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∀wal 1240 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-4 1397 ax-i5r 1425 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: (None) |
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