Theorem sbco2d 914

Description: A composition law for substitution.

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 781	. . . 4 ⊢ ((φ → ψ) → ∀z(φ → ψ))
4	3	sbco2 913	. . 3 ⊢ ([y / z][z / x](φ → ψ) ↔ [y / x](φ → ψ))
5		sbco2d.1	. . . . . 6 ⊢ (φ → ∀xφ)
6	5	sb19.21 888	. . . . 5 ⊢ ([z / x](φ → ψ) ↔ (φ → [z / x]ψ))
7	6	bisb 855	. . . 4 ⊢ ([y / z][z / x](φ → ψ) ↔ [y / z](φ → [z / x]ψ))
8	1	sb19.21 888	. . . 4 ⊢ ([y / z](φ → [z / x]ψ) ↔ (φ → [y / z][z / x]ψ))
9	7, 8	bitr 151	. . 3 ⊢ ([y / z][z / x](φ → ψ) ↔ (φ → [y / z][z / x]ψ))
10	5	sb19.21 888	. . 3 ⊢ ([y / x](φ → ψ) ↔ (φ → [y / x]ψ))
11	4, 9, 10	3bitr3 156	. 2 ⊢ ((φ → [y / z][z / x]ψ) ↔ (φ → [y / x]ψ))
12	11	pm5.74ri 445	1 ⊢ (φ → ([y / z][z / x]ψ ↔ [y / x]ψ))

Syntax hints:   → wi 2   ↔ wb 127  ∀wal 672  [wsb 852

This theorem is referenced by:  sbco3 915

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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