Theorem bnj591 35208

Description: Technical lemma for bnj852 35218. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj591.1	⊢ (𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))

Assertion

Ref	Expression
bnj591	⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))

Distinct variable groups:   𝐷,𝑗   𝜒,𝑗   𝑗,𝜒′   𝑓,𝑗   𝑔,𝑗   𝑗,𝑘   𝑗,𝑛

Allowed substitution hints:   𝜒(𝑓,𝑔,𝑘,𝑛)   𝜃(𝑓,𝑔,𝑗,𝑘,𝑛)   𝐷(𝑓,𝑔,𝑘,𝑛)   𝜒′(𝑓,𝑔,𝑘,𝑛)

Proof of Theorem bnj591

Step	Hyp	Ref	Expression
1		bnj591.1	. . 3 ⊢ (𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
2	1	sbcbii 3802	. 2 ⊢ ([𝑘 / 𝑗]𝜃 ↔ [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
3		vex 3460	. . 3 ⊢ 𝑘 ∈ V
4		fveq2 6869	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑓‘𝑗) = (𝑓‘𝑘))
5		fveq2 6869	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑔‘𝑗) = (𝑔‘𝑘))
6	4, 5	eqeq12d 2780	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑓‘𝑗) = (𝑔‘𝑗) ↔ (𝑓‘𝑘) = (𝑔‘𝑘)))
7	6	imbi2d 342	. . 3 ⊢ (𝑗 = 𝑘 → (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))))
8	3, 7	sbcie 3787	. 2 ⊢ ([𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))
9	2, 8	bitri 277	1 ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ w3a 1099   = wceq 1562   ∈ wcel 2144  [wsbc 3746  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-iota 6479  df-fv 6531

This theorem is referenced by:  bnj580  35210