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Metamath Proof Explorer

Theorem sbc3an

Description: Distribution of class substitution over triple conjunction. (Contributed by NM, 14-Dec-2006) (Revised by NM, 17-Aug-2018)

Ref Expression
Assertion sbc3an ⊢ [˙A / x]˙ φ ∧ ψ ∧ χ ↔ [˙A / x]˙ φ ∧ [˙A / x]˙ ψ ∧ [˙A / x]˙ χ

Proof

Step Hyp Ref Expression
1 sbcan ⊢ [˙A / x]˙ φ ∧ ψ ∧ χ ↔ [˙A / x]˙ φ ∧ ψ ∧ [˙A / x]˙ χ
2 sbcan ⊢ [˙A / x]˙ φ ∧ ψ ↔ [˙A / x]˙ φ ∧ [˙A / x]˙ ψ
3 1 2 bianbi ⊢ [˙A / x]˙ φ ∧ ψ ∧ χ ↔ [˙A / x]˙ φ ∧ [˙A / x]˙ ψ ∧ [˙A / x]˙ χ
4 df-3an ⊢ φ ∧ ψ ∧ χ ↔ φ ∧ ψ ∧ χ
5 4 sbcbii ⊢ [˙A / x]˙ φ ∧ ψ ∧ χ ↔ [˙A / x]˙ φ ∧ ψ ∧ χ
6 df-3an ⊢ [˙A / x]˙ φ ∧ [˙A / x]˙ ψ ∧ [˙A / x]˙ χ ↔ [˙A / x]˙ φ ∧ [˙A / x]˙ ψ ∧ [˙A / x]˙ χ
7 3 5 6 3bitr4i ⊢ [˙A / x]˙ φ ∧ ψ ∧ χ ↔ [˙A / x]˙ φ ∧ [˙A / x]˙ ψ ∧ [˙A / x]˙ χ