Theorem csbied2 3875

Description: Conversion of implicit substitution to explicit class substitution, deduction form. (Contributed by Mario Carneiro, 2-Jan-2017.)

Hypotheses

Ref	Expression
csbied2.1	⊢ (𝜑 → 𝐴 ∈ 𝑉)
csbied2.2	⊢ (𝜑 → 𝐴 = 𝐵)
csbied2.3	⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)

Assertion

Ref	Expression
csbied2	⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝐷

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem csbied2

Step	Hyp	Ref	Expression
1		csbied2.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
2		id 22	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3		csbied2.2	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	sylan9eqr 2794	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		csbied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)
6	4, 5	syldan 592	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
7	1, 6	csbied 3874	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ⦋csb 3838

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-sbc 3730  df-csb 3839

This theorem is referenced by:  prdsval  17409  cidfval  17633  monfval  17690  idfuval  17834  isnat  17908  fucco  17923  catcval  18058  xpcval  18134  1stfval  18148  2ndfval  18151  prfval  18156  evlf2  18175  curfval  18180  hofval  18209  ipoval  18487  mntoval  33057  mgcoval  33061  erlval  33334  rlocval  33335  poimirlem2  37957  rngcvalALTV  48753  ringcvalALTV  48777  upfval  49663  swapfval  49749  fucofvalg  49805  fuco21  49823  prcofvalg  49863  lanfval  50100  ranfval  50101