Theorem bnj538 31114

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 3652	. 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 196   ∈ wcel 2137  ∀wral 3048  Vcvv 3338  [wsbc 3574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-v 3340  df-sbc 3575

This theorem is referenced by:  bnj92  31237  bnj539  31266  bnj540  31267