Theorem pm11.07 2115

Description: Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)

Assertion

Ref	Expression
pm11.07	⊢ ([w / x][y / z]φ ↔ [y / x][w / z]φ)

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

Allowed substitution hint:   φ(w)

Proof of Theorem pm11.07

Step	Hyp	Ref	Expression
1		a9ev 1656	. . . . . . 7 ⊢ ∃x x = w
2		a9ev 1656	. . . . . . 7 ⊢ ∃z z = y
3	1, 2	pm3.2i 441	. . . . . 6 ⊢ (∃x x = w ∧ ∃z z = y)
4		a9ev 1656	. . . . . . 7 ⊢ ∃x x = y
5		a9ev 1656	. . . . . . 7 ⊢ ∃z z = w
6	4, 5	pm3.2i 441	. . . . . 6 ⊢ (∃x x = y ∧ ∃z z = w)
7	3, 6	2th 230	. . . . 5 ⊢ ((∃x x = w ∧ ∃z z = y) ↔ (∃x x = y ∧ ∃z z = w))
8		eeanv 1913	. . . . 5 ⊢ (∃x∃z(x = w ∧ z = y) ↔ (∃x x = w ∧ ∃z z = y))
9		eeanv 1913	. . . . 5 ⊢ (∃x∃z(x = y ∧ z = w) ↔ (∃x x = y ∧ ∃z z = w))
10	7, 8, 9	3bitr4i 268	. . . 4 ⊢ (∃x∃z(x = w ∧ z = y) ↔ ∃x∃z(x = y ∧ z = w))
11	10	anbi1i 676	. . 3 ⊢ ((∃x∃z(x = w ∧ z = y) ∧ φ) ↔ (∃x∃z(x = y ∧ z = w) ∧ φ))
12		19.41vv 1902	. . 3 ⊢ (∃x∃z((x = w ∧ z = y) ∧ φ) ↔ (∃x∃z(x = w ∧ z = y) ∧ φ))
13		19.41vv 1902	. . 3 ⊢ (∃x∃z((x = y ∧ z = w) ∧ φ) ↔ (∃x∃z(x = y ∧ z = w) ∧ φ))
14	11, 12, 13	3bitr4i 268	. 2 ⊢ (∃x∃z((x = w ∧ z = y) ∧ φ) ↔ ∃x∃z((x = y ∧ z = w) ∧ φ))
15		2sb5 2112	. 2 ⊢ ([w / x][y / z]φ ↔ ∃x∃z((x = w ∧ z = y) ∧ φ))
16		2sb5 2112	. 2 ⊢ ([y / x][w / z]φ ↔ ∃x∃z((x = y ∧ z = w) ∧ φ))
17	14, 15, 16	3bitr4i 268	1 ⊢ ([w / x][y / z]φ ↔ [y / x][w / z]φ)

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  [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: (None)