Theorem albidh 1590

Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
albidh.1	⊢ (φ → ∀xφ)
albidh.2	⊢ (φ → (ψ ↔ χ))

Assertion

Ref	Expression
albidh	⊢ (φ → (∀xψ ↔ ∀xχ))

Proof of Theorem albidh

Step	Hyp	Ref	Expression
1		albidh.1	. . 3 ⊢ (φ → ∀xφ)
2		albidh.2	. . 3 ⊢ (φ → (ψ ↔ χ))
3	1, 2	alrimih 1565	. 2 ⊢ (φ → ∀x(ψ ↔ χ))
4		albi 1564	. 2 ⊢ (∀x(ψ ↔ χ) → (∀xψ ↔ ∀xχ))
5	3, 4	syl 15	1 ⊢ (φ → (∀xψ ↔ ∀xχ))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176  ∀wal 1540

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557

This proof depends on definitions:  df-bi 177

This theorem is used by:  albidv  1625  albid  1772  ax10lem4  1941  ax9  1949  dral2  1966  dral2-o  2181  ax11indalem  2197  ax11inda2ALT  2198  ax11inda  2200