Theorem bnj591 34756

Description: Technical lemma for bnj852 34766. 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 3837	. 2 ⊢ ([𝑘 / 𝑗]𝜃 ↔ [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
3		vex 3466	. . 3 ⊢ 𝑘 ∈ V
4		fveq2 6901	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑓‘𝑗) = (𝑓‘𝑘))
5		fveq2 6901	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑔‘𝑗) = (𝑔‘𝑘))
6	4, 5	eqeq12d 2742	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑓‘𝑗) = (𝑔‘𝑗) ↔ (𝑓‘𝑘) = (𝑔‘𝑘)))
7	6	imbi2d 339	. . 3 ⊢ (𝑗 = 𝑘 → (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))))
8	3, 7	sbcie 3820	. 2 ⊢ ([𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))
9	2, 8	bitri 274	1 ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  [wsbc 3776  ‘cfv 6554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-iota 6506  df-fv 6562

This theorem is referenced by:  bnj580  34758