Theorem sbco3 2544

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 2403. (Contributed by NM, 2-Jun-1993.) (Proof shortened by Wolf Lammen, 18-Sep-2018.) (New usage is discouraged.)

Assertion

Ref	Expression
sbco3	⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Proof of Theorem sbco3

Step	Hyp	Ref	Expression
1		drsb1 2526	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
2		nfae 2464	. . . 4 ⊢ Ⅎ𝑥∀𝑥 𝑥 = 𝑦
3		sbequ12a 2289	. . . . 5 ⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
4	3	sps 2220	. . . 4 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑))
5	2, 4	sbbid 2281	. . 3 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑))
6	1, 5	bitr3d 283	. 2 ⊢ (∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑))
7		nfnae 2465	. . . 4 ⊢ Ⅎ𝑦 ¬ ∀𝑥 𝑥 = 𝑦
8		nfnae 2465	. . . 4 ⊢ Ⅎ𝑥 ¬ ∀𝑥 𝑥 = 𝑦
9		nfsb2 2514	. . . 4 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥[𝑦 / 𝑥]𝜑)
10	7, 8, 9	sbco2d 2543	. . 3 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑦][𝑦 / 𝑥]𝜑))
11		sbco 2538	. . . 4 ⊢ ([𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑥 / 𝑦]𝜑)
12	11	sbbii 2109	. . 3 ⊢ ([𝑧 / 𝑥][𝑥 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)
13	10, 12	bitr3di 288	. 2 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑))
14	6, 13	pm2.61i 183	1 ⊢ ([𝑧 / 𝑦][𝑦 / 𝑥]𝜑 ↔ [𝑧 / 𝑥][𝑥 / 𝑦]𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1558  [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:  sbcom  2545