Theorem bnj525 35235

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj525.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bnj525	⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bnj525

Step	Hyp	Ref	Expression
1		bnj525.1	. 2 ⊢ 𝐴 ∈ V
2		sbcg 3814	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  Vcvv 3453  [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 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2741  df-clel 2837  df-sbc 3743

This theorem is used by:  bnj976  35274  bnj91  35357  bnj92  35358  bnj523  35383  bnj539  35387  bnj540  35388  bnj1040  35468