Theorem csbied2 3887

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 2788	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		csbied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)
6	4, 5	syldan 591	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
7	1, 6	csbied 3886	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ⦋csb 3850

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 3742  df-csb 3851

This theorem is referenced by:  prdsval  17359  cidfval  17582  monfval  17639  idfuval  17783  isnat  17857  fucco  17872  catcval  18007  xpcval  18083  1stfval  18097  2ndfval  18100  prfval  18105  evlf2  18124  curfval  18129  hofval  18158  ipoval  18436  mntoval  32961  mgcoval  32965  erlval  33223  rlocval  33224  poimirlem2  37668  rngcvalALTV  48302  ringcvalALTV  48326  upfval  49214  swapfval  49300  fucofvalg  49356  fuco21  49374  prcofvalg  49414  lanfval  49651  ranfval  49652