Theorem cbvrexsv 3358

Description: Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2406. Use the weaker cbvrexsvw 3319 when possible. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)

Assertion

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

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

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem cbvrexsv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1947	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2194	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2289	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvrex 3354	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1947	. . . 4 ⊢ Ⅎ𝑦𝜑
6	5	nfsb 2557	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
7		nfv 1947	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
8		sbequ 2120	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
9	6, 7, 8	cbvrex 3354	. 2 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)
10	4, 9	bitri 278	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ∃𝑦 ∈ 𝐴 [𝑦 / 𝑥]𝜑)

This proof depends on syntax axioms:   ↔ wb 209  [wsb 2099  ∃wrex 3091

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-10 2179  ax-11 2195  ax-12 2216  ax-13 2406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092

This theorem is used by:  cbvexsv  45316