Theorem cbvexv 2409

Description: Rule used to change bound variables, using implicit substitution. Usage of this theorem is discouraged because it depends on ax-13 2380. See cbvexvw 2044 for a version requiring fewer axioms, to be preferred when sufficient. (Contributed by NM, 21-Jun-1993.) Remove dependency on ax-10 2152 and shorten proof. (Revised by Wolf Lammen, 11-Sep-2023.) (New usage is discouraged.)

Hypothesis

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

Assertion

Ref	Expression
cbvexv	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Distinct variable groups:   𝜑,𝑦   𝜓,𝑥

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

Proof of Theorem cbvexv

Step	Hyp	Ref	Expression
1		nfv 1921	. 2 ⊢ Ⅎ𝑦𝜑
2		nfv 1921	. 2 ⊢ Ⅎ𝑥𝜓
3		cbvalv.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
4	1, 2, 3	cbvex 2407	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 207  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-12 2189  ax-13 2380

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791

This theorem is referenced by:  cbvex2vv  2422