Theorem hband 788

Description: Deduction form of bound-variable hypothesis builder hban 704.

Hypotheses

Ref	Expression
hband.1	⊢ (φ → (ψ → ∀xψ))
hband.2	⊢ (φ → (χ → ∀xχ))

Assertion

Ref	Expression
hband	⊢ (φ → ((ψ ∧ χ) → ∀x(ψ ∧ χ)))

Proof of Theorem hband

Step	Hyp	Ref	Expression
1		hband.1	. . 3 ⊢ (φ → (ψ → ∀xψ))
2		hband.2	. . 3 ⊢ (φ → (χ → ∀xχ))
3	1, 2	anim12d 431	. 2 ⊢ (φ → ((ψ ∧ χ) → (∀xψ ∧ ∀xχ)))
4		19.26 749	. 2 ⊢ (∀x(ψ ∧ χ) ↔ (∀xψ ∧ ∀xχ))
5	3, 4	syl6ibr 186	1 ⊢ (φ → ((ψ ∧ χ) → ∀x(ψ ∧ χ)))

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

This theorem is referenced by:  axrepndlem1 3738  axrepndlem2 3739  axunndlem1 3741  axunnd 3742  axregndlem2 3749  axinfndlem1 3751  axinfnd 3752  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-gen 677

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

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