Theorem 2sb6rf 1983

Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.)

Hypotheses

Ref	Expression
2sb5rf.1	⊢ (𝜑 → ∀𝑧𝜑)
2sb5rf.2	⊢ (𝜑 → ∀𝑤𝜑)

Assertion

Ref	Expression
2sb6rf	⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Distinct variable groups:   𝑥,𝑦   𝑥,𝑤   𝑦,𝑧   𝑧,𝑤

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb6rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 ⊢ (𝜑 → ∀𝑧𝜑)
2	1	sb6rf 1846	. 2 ⊢ (𝜑 ↔ ∀𝑧(𝑧 = 𝑥 → [𝑧 / 𝑥]𝜑))
3		19.21v 1866	. . . 4 ⊢ (∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
4		sbcom2 1980	. . . . . . 7 ⊢ ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)
5	4	imbi2i 225	. . . . . 6 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
6		impexp 261	. . . . . 6 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
7	5, 6	bitri 183	. . . . 5 ⊢ (((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ (𝑧 = 𝑥 → (𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
8	7	albii 1463	. . . 4 ⊢ (∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) ↔ ∀𝑤(𝑧 = 𝑥 → (𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
9		2sb5rf.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑤𝜑)
10	9	hbsbv 1934	. . . . . 6 ⊢ ([𝑧 / 𝑥]𝜑 → ∀𝑤[𝑧 / 𝑥]𝜑)
11	10	sb6rf 1846	. . . . 5 ⊢ ([𝑧 / 𝑥]𝜑 ↔ ∀𝑤(𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑))
12	11	imbi2i 225	. . . 4 ⊢ ((𝑧 = 𝑥 → [𝑧 / 𝑥]𝜑) ↔ (𝑧 = 𝑥 → ∀𝑤(𝑤 = 𝑦 → [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)))
13	3, 8, 12	3bitr4ri 212	. . 3 ⊢ ((𝑧 = 𝑥 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
14	13	albii 1463	. 2 ⊢ (∀𝑧(𝑧 = 𝑥 → [𝑧 / 𝑥]𝜑) ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
15	2, 14	bitri 183	1 ⊢ (𝜑 ↔ ∀𝑧∀𝑤((𝑧 = 𝑥 ∧ 𝑤 = 𝑦) → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346  [wsb 1755

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756

This theorem is referenced by: (None)