Theorem bnj591 30981

Description: Technical lemma for bnj852 30991. 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 3491	. 2 ⊢ ([𝑘 / 𝑗]𝜃 ↔ [𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)))
3		vex 3203	. . 3 ⊢ 𝑘 ∈ V
4		fveq2 6191	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑓‘𝑗) = (𝑓‘𝑘))
5		fveq2 6191	. . . . 5 ⊢ (𝑗 = 𝑘 → (𝑔‘𝑗) = (𝑔‘𝑘))
6	4, 5	eqeq12d 2637	. . . 4 ⊢ (𝑗 = 𝑘 → ((𝑓‘𝑗) = (𝑔‘𝑗) ↔ (𝑓‘𝑘) = (𝑔‘𝑘)))
7	6	imbi2d 330	. . 3 ⊢ (𝑗 = 𝑘 → (((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘))))
8	3, 7	sbcie 3470	. 2 ⊢ ([𝑘 / 𝑗]((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑗) = (𝑔‘𝑗)) ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))
9	2, 8	bitri 264	1 ⊢ ([𝑘 / 𝑗]𝜃 ↔ ((𝑛 ∈ 𝐷 ∧ 𝜒 ∧ 𝜒′) → (𝑓‘𝑘) = (𝑔‘𝑘)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  [wsbc 3435  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj580  30983