Theorem gm-sbtru 36001

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 3774	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊤ ↔ ⊤))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)

Syntax hints:   ↔ wb 209  ⊤wtru 1544   ∈ wcel 2110  Vcvv 3408  [wsbc 3694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112

This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1788  df-sb 2071  df-clab 2715  df-clel 2816  df-sbc 3695

This theorem is referenced by: (None)