Theorem 2albidv 1627

Description: Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)

Hypothesis

Ref	Expression
2albidv.1	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
2albidv	⊢ (φ → (∀x∀yψ ↔ ∀x∀yχ))

Distinct variable groups:   φ,x   φ,y

Allowed substitution hints:   ψ(x,y)   χ(x,y)

Proof of Theorem 2albidv

Step	Hyp	Ref	Expression
1		2albidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	albidv 1625	. 2 ⊢ (φ → (∀yψ ↔ ∀yχ))
3	2	albidv 1625	1 ⊢ (φ → (∀x∀yψ ↔ ∀x∀yχ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616

This theorem depends on definitions:  df-bi 177

This theorem is referenced by:  2mo  2282  2eu6  2289  nnsucelr  4429  ssfin  4471  ncfinlower  4484  dff13  5472  fnfrec  6321