Theorem sbcom2 2114

Description: Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)

Assertion

Ref	Expression
sbcom2	⊢ ([w / z][y / x]φ ↔ [y / x][w / z]φ)

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

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

Proof of Theorem sbcom2

Step	Hyp	Ref	Expression
1		alcom 1737	. . . . . 6 ⊢ (∀z∀x(x = y → (z = w → φ)) ↔ ∀x∀z(x = y → (z = w → φ)))
2		bi2.04 350	. . . . . . . . 9 ⊢ ((x = y → (z = w → φ)) ↔ (z = w → (x = y → φ)))
3	2	albii 1566	. . . . . . . 8 ⊢ (∀x(x = y → (z = w → φ)) ↔ ∀x(z = w → (x = y → φ)))
4		19.21v 1890	. . . . . . . 8 ⊢ (∀x(z = w → (x = y → φ)) ↔ (z = w → ∀x(x = y → φ)))
5	3, 4	bitri 240	. . . . . . 7 ⊢ (∀x(x = y → (z = w → φ)) ↔ (z = w → ∀x(x = y → φ)))
6	5	albii 1566	. . . . . 6 ⊢ (∀z∀x(x = y → (z = w → φ)) ↔ ∀z(z = w → ∀x(x = y → φ)))
7		19.21v 1890	. . . . . . 7 ⊢ (∀z(x = y → (z = w → φ)) ↔ (x = y → ∀z(z = w → φ)))
8	7	albii 1566	. . . . . 6 ⊢ (∀x∀z(x = y → (z = w → φ)) ↔ ∀x(x = y → ∀z(z = w → φ)))
9	1, 6, 8	3bitr3i 266	. . . . 5 ⊢ (∀z(z = w → ∀x(x = y → φ)) ↔ ∀x(x = y → ∀z(z = w → φ)))
10	9	a1i 10	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → (∀z(z = w → ∀x(x = y → φ)) ↔ ∀x(x = y → ∀z(z = w → φ))))
11		sb4b 2054	. . . . 5 ⊢ (¬ ∀z z = w → ([w / z][y / x]φ ↔ ∀z(z = w → [y / x]φ)))
12		sb4b 2054	. . . . . . 7 ⊢ (¬ ∀x x = y → ([y / x]φ ↔ ∀x(x = y → φ)))
13	12	imbi2d 307	. . . . . 6 ⊢ (¬ ∀x x = y → ((z = w → [y / x]φ) ↔ (z = w → ∀x(x = y → φ))))
14	13	albidv 1625	. . . . 5 ⊢ (¬ ∀x x = y → (∀z(z = w → [y / x]φ) ↔ ∀z(z = w → ∀x(x = y → φ))))
15	11, 14	sylan9bbr 681	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → ([w / z][y / x]φ ↔ ∀z(z = w → ∀x(x = y → φ))))
16		sb4b 2054	. . . . 5 ⊢ (¬ ∀x x = y → ([y / x][w / z]φ ↔ ∀x(x = y → [w / z]φ)))
17		sb4b 2054	. . . . . . 7 ⊢ (¬ ∀z z = w → ([w / z]φ ↔ ∀z(z = w → φ)))
18	17	imbi2d 307	. . . . . 6 ⊢ (¬ ∀z z = w → ((x = y → [w / z]φ) ↔ (x = y → ∀z(z = w → φ))))
19	18	albidv 1625	. . . . 5 ⊢ (¬ ∀z z = w → (∀x(x = y → [w / z]φ) ↔ ∀x(x = y → ∀z(z = w → φ))))
20	16, 19	sylan9bb 680	. . . 4 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → ([y / x][w / z]φ ↔ ∀x(x = y → ∀z(z = w → φ))))
21	10, 15, 20	3bitr4d 276	. . 3 ⊢ ((¬ ∀x x = y ∧ ¬ ∀z z = w) → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
22	21	ex 423	. 2 ⊢ (¬ ∀x x = y → (¬ ∀z z = w → ([w / z][y / x]φ ↔ [y / x][w / z]φ)))
23		nfae 1954	. . . 4 ⊢ Ⅎz∀x x = y
24		sbequ12 1919	. . . . 5 ⊢ (x = y → (φ ↔ [y / x]φ))
25	24	sps 1754	. . . 4 ⊢ (∀x x = y → (φ ↔ [y / x]φ))
26	23, 25	sbbid 2078	. . 3 ⊢ (∀x x = y → ([w / z]φ ↔ [w / z][y / x]φ))
27		sbequ12 1919	. . . 4 ⊢ (x = y → ([w / z]φ ↔ [y / x][w / z]φ))
28	27	sps 1754	. . 3 ⊢ (∀x x = y → ([w / z]φ ↔ [y / x][w / z]φ))
29	26, 28	bitr3d 246	. 2 ⊢ (∀x x = y → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
30		sbequ12 1919	. . . 4 ⊢ (z = w → ([y / x]φ ↔ [w / z][y / x]φ))
31	30	sps 1754	. . 3 ⊢ (∀z z = w → ([y / x]φ ↔ [w / z][y / x]φ))
32		nfae 1954	. . . 4 ⊢ Ⅎx∀z z = w
33		sbequ12 1919	. . . . 5 ⊢ (z = w → (φ ↔ [w / z]φ))
34	33	sps 1754	. . . 4 ⊢ (∀z z = w → (φ ↔ [w / z]φ))
35	32, 34	sbbid 2078	. . 3 ⊢ (∀z z = w → ([y / x]φ ↔ [y / x][w / z]φ))
36	31, 35	bitr3d 246	. 2 ⊢ (∀z z = w → ([w / z][y / x]φ ↔ [y / x][w / z]φ))
37	22, 29, 36	pm2.61ii 157	1 ⊢ ([w / z][y / x]φ ↔ [y / x][w / z]φ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  2sb5rf  2117  2sb6rf  2118  2eu6  2289