Theorem cbvralsv 2712

Description: Change bound variable by using a substitution. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
cbvralsv	⊢ (∀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 cbvralsv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1521	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 1932	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 1764	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvral 2692	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1521	. . . 4 ⊢ Ⅎ𝑦𝜑
6	5	nfsb 1939	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
7		nfv 1521	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
8		sbequ 1833	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
9	6, 7, 8	cbvral 2692	. 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 1755  ∀wral 2448

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453

This theorem is referenced by:  sbralie  2714  rspsbc  3037  ralxpf  4757