Theorem bnj525 34752

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 3863	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 206   ∈ wcel 2108  Vcvv 3480  [wsbc 3788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-clel 2816  df-sbc 3789

This theorem is referenced by:  bnj976  34791  bnj91  34875  bnj92  34876  bnj523  34901  bnj539  34905  bnj540  34906  bnj1040  34986