Theorem bnj525 33738

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

Syntax hints:   ↔ wb 205   ∈ wcel 2107  Vcvv 3475  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1783  df-sb 2069  df-clab 2711  df-clel 2811  df-sbc 3778

This theorem is referenced by:  bnj976  33777  bnj91  33861  bnj92  33862  bnj523  33887  bnj539  33891  bnj540  33892  bnj1040  33972