Theorem sbco2 2542

Description: A composition law for substitution. For versions requiring fewer axioms, but more disjoint variable conditions, see sbco2v 2363 and sbco2vv 2133. Usage of this theorem is discouraged because it depends on ax-13 2403. (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 2286	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑧][𝑧 / 𝑥]𝜑))
2		sbequ 2116	. . . 4 ⊢ (𝑧 = 𝑦 → ([𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
3	1, 2	bitr3d 283	. . 3 ⊢ (𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
4	3	sps 2220	. 2 ⊢ (∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
5		nfnae 2465	. . 3 ⊢ Ⅎ𝑧 ¬ ∀𝑧 𝑧 = 𝑦
6		sbco2.1	. . . 4 ⊢ Ⅎ𝑧𝜑
7	6	nfsb4 2531	. . 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 2534	. 2 ⊢ (¬ ∀𝑧 𝑧 = 𝑦 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑))
10	4, 9	pm2.61i 183	1 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  ∀wal 1558  Ⅎwnf 1803  [wsb 2090

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-10 2175  ax-11 2191  ax-12 2212  ax-13 2403

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-ex 1800  df-nf 1804  df-sb 2091

This theorem is referenced by:  sbco2d  2543  sb7f  2556  cbvab  2834  clelsb1f  2929  sbcco  3770