Theorem cbvrex 2833

Description: Rule used to change bound variables, using implicit substitution. (Contributed by NM, 31-Jul-2003.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypotheses

Ref	Expression
cbvral.1	⊢ Ⅎyφ
cbvral.2	⊢ Ⅎxψ
cbvral.3	⊢ (x = y → (φ ↔ ψ))

Assertion

Ref	Expression
cbvrex	⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)

Distinct variable groups:   x,A   y,A

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

Proof of Theorem cbvrex

Step	Hyp	Ref	Expression
1		nfcv 2490	. 2 ⊢ ℲxA
2		nfcv 2490	. 2 ⊢ ℲyA
3		cbvral.1	. 2 ⊢ Ⅎyφ
4		cbvral.2	. 2 ⊢ Ⅎxψ
5		cbvral.3	. 2 ⊢ (x = y → (φ ↔ ψ))
6	1, 2, 3, 4, 5	cbvrexf 2831	1 ⊢ (∃x ∈ A φ ↔ ∃y ∈ A ψ)

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  ∃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:  cbvrmo  2835  cbvrexv  2837  cbvrexsv  2848  cbviun  4004