Theorem 2sb5rf 2117

Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
2sb5rf.1	⊢ Ⅎzφ
2sb5rf.2	⊢ Ⅎwφ

Assertion

Ref	Expression
2sb5rf	⊢ (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 2sb5rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 ⊢ Ⅎzφ
2	1	sb5rf 2090	. 2 ⊢ (φ ↔ ∃z(z = x ∧ [z / x]φ))
3		19.42v 1905	. . . 4 ⊢ (∃w(z = x ∧ (w = y ∧ [w / y][z / x]φ)) ↔ (z = x ∧ ∃w(w = y ∧ [w / y][z / x]φ)))
4		sbcom2 2114	. . . . . . 7 ⊢ ([z / x][w / y]φ ↔ [w / y][z / x]φ)
5	4	anbi2i 675	. . . . . 6 ⊢ (((z = x ∧ w = y) ∧ [z / x][w / y]φ) ↔ ((z = x ∧ w = y) ∧ [w / y][z / x]φ))
6		anass 630	. . . . . 6 ⊢ (((z = x ∧ w = y) ∧ [w / y][z / x]φ) ↔ (z = x ∧ (w = y ∧ [w / y][z / x]φ)))
7	5, 6	bitri 240	. . . . 5 ⊢ (((z = x ∧ w = y) ∧ [z / x][w / y]φ) ↔ (z = x ∧ (w = y ∧ [w / y][z / x]φ)))
8	7	exbii 1582	. . . 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 ⊢ Ⅎwφ
10	9	nfsb 2109	. . . . . 6 ⊢ Ⅎw[z / x]φ
11	10	sb5rf 2090	. . . . 5 ⊢ ([z / x]φ ↔ ∃w(w = y ∧ [w / y][z / x]φ))
12	11	anbi2i 675	. . . 4 ⊢ ((z = x ∧ [z / x]φ) ↔ (z = x ∧ ∃w(w = y ∧ [w / y][z / x]φ)))
13	3, 8, 12	3bitr4ri 269	. . 3 ⊢ ((z = x ∧ [z / x]φ) ↔ ∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ))
14	13	exbii 1582	. 2 ⊢ (∃z(z = x ∧ [z / x]φ) ↔ ∃z∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ))
15	2, 14	bitri 240	1 ⊢ (φ ↔ ∃z∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544  [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)