Theorem csbied2 3884

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 23	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3		csbied2.2	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	sylan9eqr 2817	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		csbied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)
6	4, 5	syldan 603	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
7	1, 6	csbied 3883	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ⦋csb 3847

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-sbc 3740  df-csb 3848

This theorem is used by:  prdsval  17540  cidfval  17764  monfval  17821  idfuval  17965  isnat  18039  fucco  18054  catcval  18189  xpcval  18265  1stfval  18279  2ndfval  18282  prfval  18287  evlf2  18306  curfval  18311  hofval  18340  ipoval  18618  angmgmval  29273  mntoval  33422  mgcoval  33426  erlval  33698  rlocval  33699  poimirlem2  38371  rngcvalALTV  49180  ringcvalALTV  49204  upfval  50102  swapfval  50188  fucofvalg  50244  fuco21  50262  prcofvalg  50302  lanfval  50539  ranfval  50540