Theorem cbvrexv 2837

Description: Change the bound variable of a restricted existential quantifier using implicit substitution. (Contributed by NM, 2-Jun-1998.)

Hypothesis

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

Assertion

Ref	Expression
cbvrexv	⊢ (∃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 cbvrexv

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	cbvrex 2833	1 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176  ∃wrex 2616

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  df-rex 2621

This theorem is referenced by:  cbvrex2v  2845  reu7  3032  ncfinraise  4482  ncfinlower  4484  nnpw1ex  4485  nnpweq  4524  vfinspss  4552  funcnvuni  5162  fun11iun  5306  clos1basesuc  5883  ncspw1eu  6160  nncdiv3  6278