Theorem cbvralv 2836

Description: Change the bound variable of a restricted universal quantifier using implicit substitution. (Contributed by NM, 28-Jan-1997.)

Hypothesis

Ref	Expression
cbvralv.1	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvralv	⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

Distinct variable groups:   x,A   y,A   φ,y   ψ,x

Allowed substitution hints:   φ(x)   ψ(y)

Proof of Theorem cbvralv

Step	Hyp	Ref	Expression
1		nfv 1619	. 2 ⊢ Ⅎyφ
2		nfv 1619	. 2 ⊢ Ⅎxψ
3		cbvralv.1	. 2 ⊢ (x = y → (φ ↔ ψ))
4	1, 2, 3	cbvral 2832	1 ⊢ (∀x ∈ A φ ↔ ∀y ∈ A ψ)

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-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620

This theorem is referenced by:  cbvral2v  2844  cbvral3v  2846  reu7  3032  nndisjeq  4430  evenodddisj  4517  nnadjoin  4521  tfinnn  4535  nnc3n3p1  6279  nchoicelem12  6301  nchoicelem17  6306