Theorem ralbida 1660

Description: Formula-building rule for restricted universal quantifier (deduction rule).

Hypotheses

Ref	Expression
ralbida.1	⊢ (φ → ∀xφ)
ralbida.2	⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))

Assertion

Ref	Expression
ralbida	⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Proof of Theorem ralbida

Step	Hyp	Ref	Expression
1		ralbida.1	. . 3 ⊢ (φ → ∀xφ)
2		ralbida.2	. . . 4 ⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))
3	2	pm5.74da 588	. . 3 ⊢ (φ → ((x ∈ A → ψ) ↔ (x ∈ A → χ)))
4	1, 3	albid 1106	. 2 ⊢ (φ → (∀x(x ∈ A → ψ) ↔ ∀x(x ∈ A → χ)))
5		df-ral 1652	. 2 ⊢ (∀x ∈ A ψ ↔ ∀x(x ∈ A → ψ))
6		df-ral 1652	. 2 ⊢ (∀x ∈ A χ ↔ ∀x(x ∈ A → χ))
7	4, 5, 6	3bitr4g 557	1 ⊢ (φ → (∀x ∈ A ψ ↔ ∀x ∈ A χ))

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223  ∀wal 956   ∈ wcel 960  ∀wral 1648

This theorem is referenced by:  ralbidva 1662  ralbid 1664  2ralbida 1680  r19.15 1756  iineq2 2583  mapxpen 4501  xpmapenlem5 4506  clm0 7083  clm0nns 7085  climabs0 7113

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 965  ax-4 975  ax-5o 977

This theorem depends on definitions:  df-bi 147  df-an 225  df-ral 1652

Copyright terms: Public domain