Theorem sbc6g 3834

Description: An equivalence for class substitution. (Contributed by NM, 11-Oct-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof shortened by SN, 5-Oct-2024.)

Assertion

Ref	Expression
sbc6g	⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem sbc6g

Step	Hyp	Ref	Expression
1		df-sbc 3805	. 2 ⊢ ([𝐴 / 𝑥]𝜑 ↔ 𝐴 ∈ {𝑥 ∣ 𝜑})
2		elab6g 3682	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))
3	1, 2	bitrid 283	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝐴 → 𝜑)))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1535   = wceq 1537   ∈ wcel 2108  {cab 2717  [wsbc 3804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-sbc 3805

This theorem is referenced by:  sbc6  3836  sbciegft  3842  sbciegftOLD  3843  sbccomlem  3891  ralsnsg  4692  fz1sbc  13660  rdgeqoa  37336  pm14.122a  44391