Theorem 2ralbidv 2657

Description: Formula-building rule for restricted universal quantifiers (deduction rule). (Contributed by NM, 28-Jan-2006.) (Revised by Szymon Jaroszewicz, 16-Mar-2007.)

Hypothesis

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

Assertion

Ref	Expression
2ralbidv	⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Distinct variable groups:   φ,x   φ,y

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

Proof of Theorem 2ralbidv

Step	Hyp	Ref	Expression
1		2ralbidv.1	. . 3 ⊢ (φ → (ψ ↔ χ))
2	1	ralbidv 2635	. 2 ⊢ (φ → (∀y ∈ B ψ ↔ ∀y ∈ B χ))
3	2	ralbidv 2635	1 ⊢ (φ → (∀x ∈ A ∀y ∈ B ψ ↔ ∀x ∈ A ∀y ∈ B χ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wral 2615

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  ax-9 1654  ax-8 1675  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-ral 2620

This theorem is referenced by:  cbvral3v  2846  isoeq1  5483  isoeq2  5484  isoeq3  5485  trd  5922  extd  5924  symd  5925  trrd  5926  antird  5929  antid  5930  connexrd  5931  connexd  5932  iserd  5943