Theorem sbcfal 3812

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 3748), but holds for sets by sbcg 3810. (Contributed by BJ, 11-Oct-2026.)

Assertion

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

Proof of Theorem sbcfal

Step	Hyp	Ref	Expression
1		sbcex 3748	. 2 ⊢ ([𝐴 / 𝑥]⊥ → 𝐴 ∈ V)
2		falim 1586	. 2 ⊢ (⊥ → 𝐴 ∈ V)
3		sbcg 3810	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊥ ↔ ⊥))
4	1, 2, 3	pm5.21nii 381	1 ⊢ ([𝐴 / 𝑥]⊥ ↔ ⊥)

This proof depends on syntax axioms:   ↔ wb 209  ⊥wfal 1581   ∈ wcel 2144  Vcvv 3450  [wsbc 3738

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-sbc 3739

This theorem is used by: (None)