Theorem sbralie 3184

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 3182	. . . 4 ⊢ (∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
2	1	sbbii 1887	. . 3 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ [𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑)
3		nfv 1843	. . . 4 ⊢ Ⅎ𝑦∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑
4		raleq 3138	. . . 4 ⊢ (𝑦 = 𝑥 → (∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑))
5	3, 4	sbie 2408	. . 3 ⊢ ([𝑥 / 𝑦]∀𝑧 ∈ 𝑦 [𝑧 / 𝑥]𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
6	2, 5	bitri 264	. 2 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑)
7		cbvralsv 3182	. . 3 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑)
8		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑧𝜑
9	8	sbco2 2415	. . . . 5 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)
10		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑥𝜓
11		sbralie.1	. . . . . 6 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
12	10, 11	sbie 2408	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓)
13	9, 12	bitri 264	. . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ 𝜓)
14	13	ralbii 2980	. . 3 ⊢ (∀𝑦 ∈ 𝑥 [𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
15	7, 14	bitri 264	. 2 ⊢ (∀𝑧 ∈ 𝑥 [𝑧 / 𝑥]𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)
16	6, 15	bitri 264	1 ⊢ ([𝑥 / 𝑦]∀𝑥 ∈ 𝑦 𝜑 ↔ ∀𝑦 ∈ 𝑥 𝜓)

Syntax hints:   → wi 4   ↔ wb 196  [wsb 1880  ∀wral 2912

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917

This theorem is referenced by:  tfinds2  7063