Theorem bnj125 32848

Description: Technical lemma for bnj150 32852. 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.)

Hypotheses

Ref	Expression
bnj125.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj125.2	⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
bnj125.3	⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
bnj125.4	⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}

Assertion

Ref	Expression
bnj125	⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓,𝑛   𝑓,𝐹   𝑅,𝑓,𝑛   𝑥,𝑓,𝑛

Allowed substitution hints:   𝜑(𝑥,𝑓,𝑛)   𝐴(𝑥)   𝑅(𝑥)   𝐹(𝑥,𝑛)   𝜑′(𝑥,𝑓,𝑛)   𝜑″(𝑥,𝑓,𝑛)

Proof of Theorem bnj125

Step	Hyp	Ref	Expression
1		bnj125.3	. 2 ⊢ (𝜑″ ↔ [𝐹 / 𝑓]𝜑′)
2		bnj125.2	. . . 4 ⊢ (𝜑′ ↔ [1o / 𝑛]𝜑)
3	2	sbcbii 3781	. . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1o / 𝑛]𝜑)
4		bnj125.1	. . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5		bnj105 32699	. . . . . 6 ⊢ 1o ∈ V
6	4, 5	bnj91 32837	. . . . 5 ⊢ ([1o / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7	6	sbcbii 3781	. . . 4 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ [𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
8		bnj125.4	. . . . . 6 ⊢ 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
9	8	bnj95 32840	. . . . 5 ⊢ 𝐹 ∈ V
10		fveq1 6770	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅))
11	10	eqeq1d 2742	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
12	9, 11	sbcie 3763	. . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
13	7, 12	bitri 274	. . 3 ⊢ ([𝐹 / 𝑓][1o / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
14	3, 13	bitri 274	. 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
15	1, 14	bitri 274	1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 205   = wceq 1542  [wsbc 3720  ∅c0 4262  {csn 4567  ⟨cop 4573  ‘cfv 6432  1_oc1o 8281   predc-bnj14 32663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-v 3433  df-sbc 3721  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-pw 4541  df-sn 4568  df-pr 4570  df-uni 4846  df-br 5080  df-suc 6271  df-iota 6390  df-fv 6440  df-1o 8288

This theorem is referenced by:  bnj150  32852  bnj153  32856