Theorem sbc2ie 3860

Description: Conversion of implicit substitution to explicit class substitution. (Contributed by NM, 16-Dec-2008.) (Revised by Mario Carneiro, 19-Dec-2013.) (Proof shortened by Gino Giotto, 12-Oct-2024.)

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	. . . 4 ⊢ 𝐵 ∈ V
3	2	a1i 11	. . 3 ⊢ (𝑥 = 𝐴 → 𝐵 ∈ V)
4		sbc2ie.3	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
5	3, 4	sbcied 3822	. 2 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ 𝜓))
6	1, 5	sbcie 3820	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  Vcvv 3474  [wsbc 3777

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-sbc 3778

This theorem is referenced by:  sbc3ie  3863  brfi1uzind  14458  opfi1uzind  14461  wrd2ind  14672  isprs  18249  isdrs  18253  istos  18370  issrg  20010  isslmd  32342  rexrabdioph  41522  rmydioph  41743  rmxdioph  41745  expdiophlem2  41751  2reu8i  45811