Theorem sbcom2v 2004

Description: Lemma for proving sbcom2 2006. It is the same as sbcom2 2006 but with additional distinct variable constraints on 𝑥 and 𝑦, and on 𝑤 and 𝑧. (Contributed by Jim Kingdon, 19-Feb-2018.)

Assertion

Ref	Expression
sbcom2v	⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

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

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

Proof of Theorem sbcom2v

Step	Hyp	Ref	Expression
1		alcom 1492	. . 3 ⊢ (∀𝑧∀𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ ∀𝑥∀𝑧(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)))
2		bi2.04 248	. . . . . 6 ⊢ ((𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ (𝑧 = 𝑤 → (𝑥 = 𝑦 → 𝜑)))
3	2	albii 1484	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ ∀𝑥(𝑧 = 𝑤 → (𝑥 = 𝑦 → 𝜑)))
4		19.21v 1887	. . . . 5 ⊢ (∀𝑥(𝑧 = 𝑤 → (𝑥 = 𝑦 → 𝜑)) ↔ (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
5	3, 4	bitri 184	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
6	5	albii 1484	. . 3 ⊢ (∀𝑧∀𝑥(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
7		19.21v 1887	. . . 4 ⊢ (∀𝑧(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
8	7	albii 1484	. . 3 ⊢ (∀𝑥∀𝑧(𝑥 = 𝑦 → (𝑧 = 𝑤 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
9	1, 6, 8	3bitr3i 210	. 2 ⊢ (∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
10		sb6 1901	. . 3 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑤 → [𝑦 / 𝑥]𝜑))
11		sb6 1901	. . . . 5 ⊢ ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → 𝜑))
12	11	imbi2i 226	. . . 4 ⊢ ((𝑧 = 𝑤 → [𝑦 / 𝑥]𝜑) ↔ (𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
13	12	albii 1484	. . 3 ⊢ (∀𝑧(𝑧 = 𝑤 → [𝑦 / 𝑥]𝜑) ↔ ∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
14	10, 13	bitri 184	. 2 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ ∀𝑧(𝑧 = 𝑤 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
15		sb6 1901	. . 3 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → [𝑤 / 𝑧]𝜑))
16		sb6 1901	. . . . 5 ⊢ ([𝑤 / 𝑧]𝜑 ↔ ∀𝑧(𝑧 = 𝑤 → 𝜑))
17	16	imbi2i 226	. . . 4 ⊢ ((𝑥 = 𝑦 → [𝑤 / 𝑧]𝜑) ↔ (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
18	17	albii 1484	. . 3 ⊢ (∀𝑥(𝑥 = 𝑦 → [𝑤 / 𝑧]𝜑) ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
19	15, 18	bitri 184	. 2 ⊢ ([𝑦 / 𝑥][𝑤 / 𝑧]𝜑 ↔ ∀𝑥(𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑤 → 𝜑)))
20	9, 14, 19	3bitr4i 212	1 ⊢ ([𝑤 / 𝑧][𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥][𝑤 / 𝑧]𝜑)

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362  [wsb 1776

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-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-11 1520  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-sb 1777

This theorem is referenced by:  sbcom2v2  2005