Theorem sbco2d 1840

Description: A composition law for substitution. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
sbco2d.1	⊢ (𝜑 → ∀𝑥𝜑)
sbco2d.2	⊢ (𝜑 → ∀𝑧𝜑)
sbco2d.3	⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))

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 ⊢ (𝜑 → ∀𝑧𝜑)
2		sbco2d.3	. . . . 5 ⊢ (𝜑 → (𝜓 → ∀𝑧𝜓))
3	1, 2	hbim1 1462	. . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑧(𝜑 → 𝜓))
4	3	sbco2h 1838	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
5		sbco2d.1	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑)
6	5	sbrim 1830	. . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
7	6	sbbii 1648	. . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
8	1	sbrim 1830	. . . 4 ⊢ ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
9	7, 8	bitri 173	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
10	5	sbrim 1830	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
11	4, 9, 10	3bitr3i 199	. 2 ⊢ ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
12	11	pm5.74ri 170	1 ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 98  ∀wal 1241  [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)