Theorem albidh 1502

Description: Formula-building rule for universal quantifier (deduction form). (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 1491	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
4		albi 1490	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
5	3, 4	syl 14	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1469  ax-gen 1471

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfbidf  1561  albid  1637  dral2  1753  ax11v2  1842  albidv  1846  equs5or  1852  sbal2  2047  eubidh  2059