Theorem sbralie 2604

Description: Implicit to explicit substitution that swaps variables in a quantified expression. (Contributed by NM, 5-Sep-2004.)

Hypothesis

Ref	Expression
sbralie.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbralie	⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)

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

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem sbralie

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cbvralsv 2602	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
2	1	sbbii 1696	. . 3 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
3		nfv 1467	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑
4		raleq 2563	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑))
5	3, 4	sbie 1722	. . 3 ⊢ ([𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
6	2, 5	bitri 183	. 2 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
7		cbvralsv 2602	. . 3 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
8		nfv 1467	. . . . . 6 ⊢ Ⅎ𝑧𝜑
9	8	sbco2 1888	. . . . 5 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
10		nfv 1467	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11		sbralie.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
12	10, 11	sbie 1722	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
13	9, 12	bitri 183	. . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ 𝜓)
14	13	ralbii 2385	. . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
15	7, 14	bitri 183	. 2 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
16	6, 15	bitri 183	1 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 104  [wsb 1693  ∀wral 2360

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071

This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365

This theorem is referenced by: (None)