Theorem bnj591 35308

Description: Technical lemma for bnj852 35318. 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 3799	. 2 ⊢ ([𝑘 / 𝑗]𝜃 ↔ [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
3		vex 3458	. . 3 ⊢ 𝑘 ∈ V
4		fveq2 6881	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑓‘𝑗) = (𝑓‘𝑘))
5		fveq2 6881	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑔‘𝑗) = (𝑔‘𝑘))
6	4, 5	eqeq12d 2778	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑓‘𝑗) = (𝑔‘𝑗) ↔ (𝑓‘𝑘) = (𝑔‘𝑘)))
7	6	imbi2d 343	. . 3 ⊢ (𝑗 = 𝑘 → (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))))
8	3, 7	sbcie 3784	. 2 ⊢ ([𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))
9	2, 8	bitri 278	1 ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  [wsbc 3743  ‘cfv 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-iota 6492  df-fv 6544

This theorem is used by:  bnj580  35310