Theorem sbco2d 1954

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 1558	. . . 4 ⊢ ((𝜑 → 𝜓) → ∀𝑧(𝜑 → 𝜓))
4	3	sbco2h 1952	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑥](𝜑 → 𝜓))
5		sbco2d.1	. . . . . 6 ⊢ (𝜑 → ∀𝑥𝜑)
6	5	sbrim 1944	. . . . 5 ⊢ ([𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑧 / 𝑥]𝜓))
7	6	sbbii 1753	. . . 4 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ [𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓))
8	1	sbrim 1944	. . . 4 ⊢ ([𝑦 / 𝑧](𝜑 → [𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
9	7, 8	bitri 183	. . 3 ⊢ ([𝑦 / 𝑧][𝑧 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓))
10	5	sbrim 1944	. . 3 ⊢ ([𝑦 / 𝑥](𝜑 → 𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
11	4, 9, 10	3bitr3i 209	. 2 ⊢ ((𝜑 → [𝑦 / 𝑧][𝑧 / 𝑥]𝜓) ↔ (𝜑 → [𝑦 / 𝑥]𝜓))
12	11	pm5.74ri 180	1 ⊢ (𝜑 → ([𝑦 / 𝑧][𝑧 / 𝑥]𝜓 ↔ [𝑦 / 𝑥]𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341  [wsb 1750

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751

This theorem is referenced by: (None)