Metamath Proof Explorer

Theorem sbco2d

Description: A composition law for substitution. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by NM, 2-Jun-1993) (Revised by Mario Carneiro, 6-Oct-2016) (New usage is discouraged.)

Ref Expression
Hypotheses sbco2d.1 ⊢ Ⅎ x φ
sbco2d.2 ⊢ Ⅎ z φ
sbco2d.3 ⊢ φ → Ⅎ z ψ
Assertion sbco2d ⊢ φ → y z z x ψ ↔ y x ψ

Proof

Step Hyp Ref Expression
1 sbco2d.1 ⊢ Ⅎ x φ
2 sbco2d.2 ⊢ Ⅎ z φ
3 sbco2d.3 ⊢ φ → Ⅎ z ψ
4 2 3 nfim1 ⊢ Ⅎ z φ → ψ
5 4 sbco2 ⊢ y z z x φ → ψ ↔ y x φ → ψ
6 1 sbrim ⊢ z x φ → ψ ↔ φ → z x ψ
7 6 sbbii ⊢ y z z x φ → ψ ↔ y z φ → z x ψ
8 2 sbrim ⊢ y z φ → z x ψ ↔ φ → y z z x ψ
9 7 8 bitri ⊢ y z z x φ → ψ ↔ φ → y z z x ψ
10 1 sbrim ⊢ y x φ → ψ ↔ φ → y x ψ
11 5 9 10 3bitr3i ⊢ φ → y z z x ψ ↔ φ → y x ψ
12 11 pm5.74ri ⊢ φ → y z z x ψ ↔ y x ψ