Theorem gm-sbtru 38783

Description: Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)

Hypothesis

Ref	Expression
gm-sbtru.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
gm-sbtru	⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)

Proof of Theorem gm-sbtru

Step	Hyp	Ref	Expression
1		gm-sbtru.1	. 2 ⊢ 𝐴 ∈ V
2		sbcg 3815	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊤ ↔ ⊤))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)

This proof depends on syntax axioms:   ↔ wb 209  ⊤wtru 1570   ∈ wcel 2142  Vcvv 3454  [wsbc 3743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144

This proof depends on definitions:  df-bi 210  df-an 401  df-ex 1809  df-sb 2096  df-clab 2741  df-clel 2837  df-sbc 3744

This theorem is used by: (None)