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	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\perp \leftrightarrow \perp$$

Proof

Step	Hyp	Ref	Expression
1		sbcex	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\perp \to A\in \mathrm{V}$$
2		falim	$$⊢\perp \to A\in \mathrm{V}$$
3		sbcg	$$⊢A\in \mathrm{V}\to \left(\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\perp \leftrightarrow \perp \right)$$
4	1 2 3	pm5.21nii	$$⊢\underset{\dot{}}{[}A/x\underset{\dot{}}{]}\perp \leftrightarrow \perp$$