Theorem sbc2ie 3812

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 GG, 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 3780	. 2 ⊢ (𝑥 = 𝐴 → ([𝐵 / 𝑦]𝜑 ↔ 𝜓))
6	1, 5	sbcie 3778	1 ⊢ ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  Vcvv 3436  [wsbc 3736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-sbc 3737

This theorem is referenced by:  sbc3ie  3814  brfi1uzind  14415  opfi1uzind  14418  wrd2ind  14630  isprs  18202  isdrs  18207  istos  18322  issrg  20106  isslmd  33171  rexrabdioph  42835  rmydioph  43055  rmxdioph  43057  expdiophlem2  43063  2reu8i  47152