Theorem sbralie 2590

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 2588	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
2	1	sbbii 1688	. . 3 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
3		nfv 1461	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑
4		raleq 2549	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑))
5	3, 4	sbie 1714	. . 3 ⊢ ([𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
6	2, 5	bitri 182	. 2 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
7		cbvralsv 2588	. . 3 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
8		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑧𝜑
9	8	sbco2 1880	. . . . 5 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
10		nfv 1461	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11		sbralie.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
12	10, 11	sbie 1714	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
13	9, 12	bitri 182	. . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ 𝜓)
14	13	ralbii 2372	. . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
15	7, 14	bitri 182	. 2 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
16	6, 15	bitri 182	1 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 103  [wsb 1685  ∀wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353

This theorem is referenced by: (None)