Theorem sbc2ie 3472

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.)

Hypotheses

Ref	Expression
sbc2ie.1	⊢ 𝐴 ∈ V
sbc2ie.2	⊢ 𝐵 ∈ V
sbc2ie.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
sbc2ie	⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵   𝜓,𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbc2ie

Step	Hyp	Ref	Expression
1		sbc2ie.1	. 2 ⊢ 𝐴 ∈ V
2		sbc2ie.2	. 2 ⊢ 𝐵 ∈ V
3		nfv 1830	. . 3 ⊢ Ⅎ𝑥𝜓
4		nfv 1830	. . 3 ⊢ Ⅎ𝑦𝜓
5	2	nfth 1718	. . 3 ⊢ Ⅎ𝑥 𝐵 ∈ V
6		sbc2ie.3	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7	3, 4, 5, 6	sbc2iegf 3471	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓))
8	1, 2, 7	mp2an 704	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173  [wsbc 3402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-v 3175  df-sbc 3403

This theorem is referenced by:  sbc3ie  3474  brfi1uzind  13135  opfi1uzind  13138  brfi1uzindOLD  13141  opfi1uzindOLD  13144  wrd2ind  13329  isprs  16753  isdrs  16757  istos  16858  issrg  18330  isslmd  29086  rexrabdioph  36376  rmydioph  36599  rmxdioph  36601  expdiophlem2  36607