Theorem bnj525 31618

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 3777	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑥]𝜑 ↔ 𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝜑)

Syntax hints:   ↔ wb 207   ∈ wcel 2080  Vcvv 3436  [wsbc 3707

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1778  ax-4 1792  ax-5 1889  ax-6 1948  ax-7 1993  ax-8 2082  ax-9 2090  ax-12 2140  ax-ext 2768

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1763  df-nf 1767  df-sb 2042  df-clab 2775  df-cleq 2787  df-clel 2862  df-sbc 3708

This theorem is referenced by:  bnj976  31658  bnj91  31741  bnj92  31742  bnj523  31767  bnj539  31771  bnj540  31772  bnj1040  31850