Theorem sbco2 2086

Description: A composition law for substitution. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sbco2.1	⊢ Ⅎzφ

Assertion

Ref	Expression
sbco2	⊢ ([y / z][z / x]φ ↔ [y / x]φ)

Proof of Theorem sbco2

Step	Hyp	Ref	Expression
1		sbco2.1	. . . . . 6 ⊢ Ⅎzφ
2	1	sbid2 2084	. . . . 5 ⊢ ([x / z][z / x]φ ↔ φ)
3		sbequ 2060	. . . . 5 ⊢ (x = y → ([x / z][z / x]φ ↔ [y / z][z / x]φ))
4	2, 3	syl5bbr 250	. . . 4 ⊢ (x = y → (φ ↔ [y / z][z / x]φ))
5		sbequ12 1919	. . . 4 ⊢ (x = y → (φ ↔ [y / x]φ))
6	4, 5	bitr3d 246	. . 3 ⊢ (x = y → ([y / z][z / x]φ ↔ [y / x]φ))
7	6	sps 1754	. 2 ⊢ (∀x x = y → ([y / z][z / x]φ ↔ [y / x]φ))
8		nfnae 1956	. . . 4 ⊢ Ⅎx ¬ ∀x x = y
9	1	nfs1 2044	. . . . 5 ⊢ Ⅎx[z / x]φ
10	9	nfsb4 2081	. . . 4 ⊢ (¬ ∀x x = y → Ⅎx[y / z][z / x]φ)
11	4	a1i 10	. . . 4 ⊢ (¬ ∀x x = y → (x = y → (φ ↔ [y / z][z / x]φ)))
12	8, 10, 11	sbied 2036	. . 3 ⊢ (¬ ∀x x = y → ([y / x]φ ↔ [y / z][z / x]φ))
13	12	bicomd 192	. 2 ⊢ (¬ ∀x x = y → ([y / z][z / x]φ ↔ [y / x]φ))
14	7, 13	pm2.61i 156	1 ⊢ ([y / z][z / x]φ ↔ [y / x]φ)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbco2d  2087  equsb3  2102  elsb1  2103  elsb2  2104  dfsb7  2119  sb7f  2120  2eu6  2289  eqsb1  2454  clelsb1  2455  clelsb2  2456  sbralie  2849  sbcco  3069