Theorem hbbid 1464

Description: Deduction form of bound-variable hypothesis builder hbbi 1437. (Contributed by NM, 1-Jan-2002.)

Hypotheses

Ref	Expression
hbbid.1	⊢ (φ → ∀xφ)
hbbid.2	⊢ (φ → (ψ → ∀xψ))
hbbid.3	⊢ (φ → (χ → ∀xχ))

Assertion

Ref	Expression
hbbid	⊢ (φ → ((ψ ↔ χ) → ∀x(ψ ↔ χ)))

Proof of Theorem hbbid

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)))

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)