Theorem bnj1464 35241

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1464.1	⊢ (𝜓 → ∀𝑥𝜓)
bnj1464.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
bnj1464	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑉

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem bnj1464

Step	Hyp	Ref	Expression
1		bnj1464.1	. . 3 ⊢ (𝜓 → ∀𝑥𝜓)
2	1	nf5i 2180	. 2 ⊢ Ⅎ𝑥𝜓
3		bnj1464.2	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4	2, 3	sbciegf 3781	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∀wal 1567   = wceq 1569   ∈ wcel 2142  [wsbc 3743

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-12 2212  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-ex 1809  df-nf 1813  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-sbc 3744

This theorem is used by:  bnj1465  35242  bnj1468  35243