Theorem csbied2 3042

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 19	. . . 4 ⊢ (𝑥 = 𝐴 → 𝑥 = 𝐴)
3		csbied2.2	. . . 4 ⊢ (𝜑 → 𝐴 = 𝐵)
4	2, 3	sylan9eqr 2192	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
5		csbied2.3	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝐵) → 𝐶 = 𝐷)
6	4, 5	syldan 280	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐷)
7	1, 6	csbied 3041	1 ⊢ (𝜑 → ⦋𝐴 / 𝑥⦌𝐶 = 𝐷)

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1331   ∈ wcel 1480  ⦋csb 2998

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-sbc 2905  df-csb 2999

This theorem is referenced by: (None)