Theorem cbvrexsv 2669

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 1508	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 1912	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 1744	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvrex 2651	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1508	. . . 4 ⊢ Ⅎ𝑦𝜑
6	5	nfsb 1919	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
7		nfv 1508	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
8		sbequ 1812	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
9	6, 7, 8	cbvrex 2651	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
10	4, 9	bitri 183	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

Syntax hints:   ↔ wb 104  [wsb 1735  ∃wrex 2417

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-cleq 2132  df-clel 2135  df-nfc 2270  df-rex 2422

This theorem is referenced by:  rspesbca  2993  rexxpf  4686