Theorem sbralie 2721

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 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

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

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 2719	. . 3 ⊢ (∀𝑦 ∈ 𝑥 𝜓 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
2	1	sbbii 1765	. 2 ⊢ ([𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓 ↔ [𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓)
3		nfv 1528	. . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓
4		raleq 2672	. . 3 ⊢ (𝑥 = 𝑦 → (∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓))
5	3, 4	sbie 1791	. 2 ⊢ ([𝑦 / 𝑥]∀𝑧 ∈ 𝑥 [𝑧 / 𝑦]𝜓 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓)
6		cbvralsv 2719	. . 3 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓)
7		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑧𝜓
8	7	sbco2 1965	. . . . 5 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ [𝑥 / 𝑦]𝜓)
9		nfv 1528	. . . . . 6 ⊢ Ⅎ𝑦𝜑
10		sbralie.1	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
11	10	bicomd 141	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜑))
12	11	equcoms 1708	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝜓 ↔ 𝜑))
13	9, 12	sbie 1791	. . . . 5 ⊢ ([𝑥 / 𝑦]𝜓 ↔ 𝜑)
14	8, 13	bitri 184	. . . 4 ⊢ ([𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ 𝜑)
15	14	ralbii 2483	. . 3 ⊢ (∀𝑥 ∈ 𝑦 [𝑥 / 𝑧][𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑)
16	6, 15	bitri 184	. 2 ⊢ (∀𝑧 ∈ 𝑦 [𝑧 / 𝑦]𝜓 ↔ ∀𝑥 ∈ 𝑦 𝜑)
17	2, 5, 16	3bitrri 207	1 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑦 / 𝑥]∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 105  [wsb 1762  ∀wral 2455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460

This theorem is referenced by: (None)