Theorem cbvrexsvw 3403

Description: Change bound variable by using a substitution. Version of cbvrexsv 3405 with a disjoint variable condition, which does not require ax-13 2372. (Contributed by NM, 2-Mar-2008.) (Revised by Gino Giotto, 10-Jan-2024.)

Assertion

Ref	Expression
cbvrexsvw	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsvw

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1917	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2153	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2244	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvrexw 3374	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1917	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
6		nfv 1917	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
7		sbequ 2086	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
8	5, 6, 7	cbvrexw 3374	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
9	4, 8	bitri 274	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 205  [wsb 2067  ∃wrex 3065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-ex 1783  df-nf 1787  df-sb 2068  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070

This theorem is referenced by:  rspesbca  3814  ac6sf  10245  ac6gf  35890