Theorem sbciegf 2892

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 14-Dec-2005.) (Revised by Mario Carneiro, 13-Oct-2016.)

Hypotheses

Ref	Expression
sbciegf.1	⊢ Ⅎ𝑥𝜓
sbciegf.2	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))

Assertion

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

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem sbciegf

Step	Hyp	Ref	Expression
1		sbciegf.1	. 2 ⊢ Ⅎ𝑥𝜓
2		sbciegf.2	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
3	2	ax-gen 1393	. 2 ⊢ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		sbciegft 2891	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ Ⅎ𝑥𝜓 ∧ ∀𝑥(𝑥 = 𝐴 → (𝜑 ↔ 𝜓))) → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))
5	1, 3, 4	mp3an23 1275	1 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝜑 ↔ 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1297   = wceq 1299  Ⅎwnf 1404   ∈ wcel 1448  [wsbc 2862

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-sbc 2863

This theorem is referenced by:  sbcieg  2893  iunxsngf  3837  opelopabf  4134  eqerlem  6390