Theorem bnj525 35097

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 3824	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 209   ∈ wcel 2150  Vcvv 3462  [wsbc 3752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1808  df-sb 2099  df-clab 2749  df-clel 2845  df-sbc 3753

This theorem is referenced by:  bnj976  35136  bnj91  35219  bnj92  35220  bnj523  35245  bnj539  35249  bnj540  35250  bnj1040  35330