Theorem bnj538 35237

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.) (Proof shortened by OpenAI, 30-Mar-2020.)

Hypothesis

Ref	Expression
bnj538.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bnj538	⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐴(𝑦)   𝐵(𝑥)

Proof of Theorem bnj538

Step	Hyp	Ref	Expression
1		bnj538.1	. 2 ⊢ 𝐴 ∈ V
2		sbcralg 3824	. 2 ⊢ (𝐴 ∈ V → ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑))
3	1, 2	ax-mp 5	1 ⊢ ([𝐴 / 𝑦]∀𝑥 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐵 [𝐴 / 𝑦]𝜑)

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2145  ∀wral 3078  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  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-v 3455  df-sbc 3743

This theorem is used by:  bnj92  35358  bnj539  35387  bnj540  35388