Theorem sbcfal

Description: Substituting a class in the false truth constant results in the false truth constant. The analogous result with the true truth constant does not hold in general since substituting a proper class results in falsehood ( sbcex ), but holds for sets by sbcg . (Contributed by BJ, 11-Oct-2026)

	Ref	Expression
Assertion	sbcfal	⊢ ( [ 𝐴 / 𝑥 ] ⊥ ↔ ⊥ )

Proof

Step	Hyp	Ref	Expression
1		sbcex	⊢ ( [ 𝐴 / 𝑥 ] ⊥ → 𝐴 ∈ V )
2		falim	⊢ ( ⊥ → 𝐴 ∈ V )
3		sbcg	⊢ ( 𝐴 ∈ V → ( [ 𝐴 / 𝑥 ] ⊥ ↔ ⊥ ) )
4	1 2 3	pm5.21nii	⊢ ( [ 𝐴 / 𝑥 ] ⊥ ↔ ⊥ )