Theorem albidh 1793

Description: Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-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 1751	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		albi 1746	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
5	3, 4	syl 17	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 196  ∀wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737

This theorem depends on definitions:  df-bi 197

This theorem is referenced by:  albidv  1849  albid  2090  albidOLD  2199  dral2-o  34215  ax12indalem  34230  ax12inda2ALT  34231  ax12inda  34233