Theorem sbcfal 3816

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 3752), but holds for sets by sbcg 3814. (Contributed by BJ, 11-Oct-2026.)

Assertion

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

Proof of Theorem sbcfal

Step	Hyp	Ref	Expression
1		sbcex 3752	. 2 ⊢ ([𝐴 / 𝑥]⊥ → 𝐴 ∈ V)
2		falim 1587	. 2 ⊢ (⊥ → 𝐴 ∈ V)
3		sbcg 3814	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊥ ↔ ⊥))
4	1, 2, 3	pm5.21nii 381	1 ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥)

This proof depends on syntax axioms:   ↔ wb 209  ⊥wfal 1582   ∈ wcel 2147  Vcvv 3454  [wsbc 3742

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-sbc 3743

This theorem is used by: (None)