Theorem sbco2d 2087

Description: A composition law for substitution. (Contributed by NM, 5-Aug-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 1811	. . . 4 ⊢ Ⅎz(φ → ψ)
4	3	sbco2 2086	. . 3 ⊢ ([y / z][z / x](φ → ψ) ↔ [y / x](φ → ψ))
5		sbco2d.1	. . . . . 6 ⊢ Ⅎxφ
6	5	sbrim 2067	. . . . 5 ⊢ ([z / x](φ → ψ) ↔ (φ → [z / x]ψ))
7	6	sbbii 1653	. . . 4 ⊢ ([y / z][z / x](φ → ψ) ↔ [y / z](φ → [z / x]ψ))
8	1	sbrim 2067	. . . 4 ⊢ ([y / z](φ → [z / x]ψ) ↔ (φ → [y / z][z / x]ψ))
9	7, 8	bitri 240	. . 3 ⊢ ([y / z][z / x](φ → ψ) ↔ (φ → [y / z][z / x]ψ))
10	5	sbrim 2067	. . 3 ⊢ ([y / x](φ → ψ) ↔ (φ → [y / x]ψ))
11	4, 9, 10	3bitr3i 266	. 2 ⊢ ((φ → [y / z][z / x]ψ) ↔ (φ → [y / x]ψ))
12	11	pm5.74ri 237	1 ⊢ (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Syntax hints:   → wi 4   ↔ wb 176  Ⅎwnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbco3  2088