Theorem gm-sbtru 37441

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 3856	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]⊤ ↔ ⊤))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]⊤ ↔ ⊤)

Syntax hints:   ↔ wb 205  ⊤wtru 1541   ∈ wcel 2105  Vcvv 3473  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1781  df-sb 2067  df-clab 2709  df-clel 2809  df-sbc 3778

This theorem is referenced by: (None)