Theorem cbvrexsv 3183

Description: Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

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

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2437	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2111	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvrex 3168	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1843	. . . 4 ⊢ Ⅎ𝑦𝜑
6	5	nfsb 2440	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
7		nfv 1843	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
8		sbequ 2376	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
9	6, 7, 8	cbvrex 3168	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
10	4, 9	bitri 264	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 196  [wsb 1880  ∃wrex 2913

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918

This theorem is referenced by:  rspesbca  3520  ac6sf  9311  ac6gf  33527  cbvexsv  38762