Theorem cbvexd 1983

Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2077. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)

Hypotheses

Ref	Expression
cbvexd.1	⊢ Ⅎ𝑦𝜑
cbvexd.2	⊢ (𝜑 → Ⅎ𝑦𝜓)
cbvexd.3	⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))

Assertion

Ref	Expression
cbvexd	⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Distinct variable groups:   𝜑,𝑥   𝜒,𝑥

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

Proof of Theorem cbvexd

Step	Hyp	Ref	Expression
1		cbvexd.1	. . 3 ⊢ Ⅎ𝑦𝜑
2	1	nfri 1572	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		cbvexd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑦𝜓)
4	3	nfrd 1573	. 2 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
5		cbvexd.3	. 2 ⊢ (𝜑 → (𝑥 = 𝑦 → (𝜓 ↔ 𝜒)))
6	2, 4, 5	cbvexdh 1982	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 105  Ⅎwnf 1513  ∃wex 1545

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-nf 1514

This theorem is referenced by:  cbvexdva  1985  vtoclgft  2873  bdsepnft  16827  strcollnft  16924