Theorem sbco2 2514

Description: A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2330 and sbco2vv 2098. Usage of this theorem is discouraged because it depends on ax-13 2375. (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 2250	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑))
2		sbequ 2082	. . . 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 2184	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
5		nfnae 2437	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
6		sbco2.1	. . . 4 ⊢ Ⅎ𝑧𝜑
7	6	nfsb4 2503	. . 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 2506	. 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 206  ∀wal 1537  Ⅎwnf 1782  [wsb 2063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-10 2140  ax-11 2156  ax-12 2176  ax-13 2375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-ex 1779  df-nf 1783  df-sb 2064

This theorem is referenced by:  sbco2d  2515  sb7f  2528  cbvab  2806  clelsb2OLD  2862  clelsb1f  2902  sbcco  3789