Theorem 2sb5rf 1862

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

Hypotheses

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

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 ⊢ (φ → ∀zφ)
2	1	sb5rf 1729	. 2 ⊢ (φ ↔ ∃z(z = x ∧ [z / x]φ))
3		19.42v 1783	. . . 4 ⊢ (∃w(z = x ∧ (w = y ∧ [w / y][z / x]φ)) ↔ (z = x ∧ ∃w(w = y ∧ [w / y][z / x]φ)))
4		sbcom2 1860	. . . . . . 7 ⊢ ([z / x][w / y]φ ↔ [w / y][z / x]φ)
5	4	anbi2i 430	. . . . . 6 ⊢ (((z = x ∧ w = y) ∧ [z / x][w / y]φ) ↔ ((z = x ∧ w = y) ∧ [w / y][z / x]φ))
6		anass 381	. . . . . 6 ⊢ (((z = x ∧ w = y) ∧ [w / y][z / x]φ) ↔ (z = x ∧ (w = y ∧ [w / y][z / x]φ)))
7	5, 6	bitri 173	. . . . 5 ⊢ (((z = x ∧ w = y) ∧ [z / x][w / y]φ) ↔ (z = x ∧ (w = y ∧ [w / y][z / x]φ)))
8	7	exbii 1493	. . . 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 1814	. . . . . 6 ⊢ ([z / x]φ → ∀w[z / x]φ)
11	10	sb5rf 1729	. . . . 5 ⊢ ([z / x]φ ↔ ∃w(w = y ∧ [w / y][z / x]φ))
12	11	anbi2i 430	. . . 4 ⊢ ((z = x ∧ [z / x]φ) ↔ (z = x ∧ ∃w(w = y ∧ [w / y][z / x]φ)))
13	3, 8, 12	3bitr4ri 202	. . 3 ⊢ ((z = x ∧ [z / x]φ) ↔ ∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ))
14	13	exbii 1493	. 2 ⊢ (∃z(z = x ∧ [z / x]φ) ↔ ∃z∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ))
15	2, 14	bitri 173	1 ⊢ (φ ↔ ∃z∃w((z = x ∧ w = y) ∧ [z / x][w / y]φ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240  ∃wex 1378  [wsb 1642

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643

This theorem is referenced by: (None)