Theorem hbbid 789

Description: Deduction form of bound-variable hypothesis builder hbbi 705.

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 787	. . 3 ⊢ (φ → ((ψ → χ) → ∀x(ψ → χ)))
5	1, 3, 2	hbimd 787	. . 3 ⊢ (φ → ((χ → ψ) → ∀x(χ → ψ)))
6	4, 5	anim12d 431	. 2 ⊢ (φ → (((ψ → χ) ∧ (χ → ψ)) → (∀x(ψ → χ) ∧ ∀x(χ → ψ))))
7		bi 396	. 2 ⊢ ((ψ ↔ χ) ↔ ((ψ → χ) ∧ (χ → ψ)))
8		albi 785	. 2 ⊢ (∀x(ψ ↔ χ) ↔ (∀x(ψ → χ) ∧ ∀x(χ → ψ)))
9	6, 7, 8	3imtr4g 426	1 ⊢ (φ → ((ψ ↔ χ) → ∀x(ψ ↔ χ)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672

This theorem is referenced by:  hbeu 1016  axextnd 3737  axrepndlem1 3738  axrepndlem2 3739  axacndlem4 3756  axacndlem5 3757  axacnd 3758

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198

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