Theorem bnj538 35099

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 3835	. 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)

Syntax hints:   ↔ wb 209   ∈ wcel 2150  ∀wral 3086  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  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1571  df-ex 1808  df-nf 1812  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-v 3464  df-sbc 3753

This theorem is referenced by:  bnj92  35220  bnj539  35249  bnj540  35250