Theorem cbvralsv 3362

Description: Change bound variable by using a substitution. Usage of this theorem is discouraged because it depends on ax-13 2410. Use the weaker cbvralsvw 3322 when possible. (Contributed by NM, 20-Nov-2005.) (Revised by Andrew Salmon, 11-Jul-2011.) (New usage is discouraged.)

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 1941	. . 3 ⊢ Ⅎ𝑧𝜑
2		nfs1v 2197	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑥]𝜑
3		sbequ12 2293	. . 3 ⊢ (𝑥 = 𝑧 → (𝜑 ↔ [𝑧 / 𝑥]𝜑))
4	1, 2, 3	cbvral 3358	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝜑 ↔ ∀𝑧 ∈ 𝐴 [𝑧 / 𝑥]𝜑)
5		nfv 1941	. . . 4 ⊢ Ⅎ𝑦𝜑
6	5	nfsb 2561	. . 3 ⊢ Ⅎ𝑦[𝑧 / 𝑥]𝜑
7		nfv 1941	. . 3 ⊢ Ⅎ𝑧[𝑦 / 𝑥]𝜑
8		sbequ 2123	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
9	6, 7, 8	cbvral 3358	. 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)

Syntax hints:   ↔ wb 209  [wsb 2097  ∀wral 3085

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-10 2182  ax-11 2198  ax-12 2219  ax-13 2410

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clel 2844  df-nfc 2918  df-ral 3086

This theorem is referenced by: (None)