Theorem sbco2 2509

Description: A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2325 and sbco2vv 2099. Usage of this theorem is discouraged because it depends on ax-13 2370. (Contributed by NM, 30-Jun-1994.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 17-Sep-2018.) (New usage is discouraged.)

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		sbequ12 2242	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑))
2		sbequ 2085	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	bitr3d 281	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	sps 2177	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
5		nfnae 2432	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
6		sbco2.1	. . . 4 ⊢ Ⅎ𝑧𝜑
7	6	nfsb4 2498	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → Ⅎ𝑧[𝑦 / 𝑥]𝜑)
8	2	a1i 11	. . 3 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)))
9	5, 7, 8	sbied 2501	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
10	4, 9	pm2.61i 182	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205  ∀wal 1538  Ⅎwnf 1784  [wsb 2066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067

This theorem is referenced by:  sbco2d  2510  sb7f  2523  cbvab  2807  clelsb2OLD  2861  clelsb1f  2907  sbcco  3803