Theorem sbco2d 2416

Description: A composition law for substitution. (Contributed by NM, 2-Jun-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
sbco2d.1	⊢ Ⅎ𝑥𝜑
sbco2d.2	⊢ Ⅎ𝑧𝜑
sbco2d.3	⊢ (𝜑 → Ⅎ𝑧𝜓)

Assertion

Ref	Expression
sbco2d	⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Proof of Theorem sbco2d

Step	Hyp	Ref	Expression
1		sbco2d.2	. . . . 5 ⊢ Ⅎ𝑧𝜑
2		sbco2d.3	. . . . 5 ⊢ (𝜑 → Ⅎ𝑧𝜓)
3	1, 2	nfim1 2067	. . . 4 ⊢ Ⅎ𝑧(𝜑 → 𝜓)
4	3	sbco2 2415	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
5		sbco2d.1	. . . . . 6 ⊢ Ⅎ𝑥𝜑
6	5	sbrim 2396	. . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
7	6	sbbii 1887	. . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
8	1	sbrim 2396	. . . 4 ⊢ ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
9	7, 8	bitri 264	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
10	5	sbrim 2396	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
11	4, 9, 10	3bitr3i 290	. 2 ⊢ ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
12	11	pm5.74ri 261	1 ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 196  Ⅎwnf 1708  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbco3  2417