Theorem bnj125 30196

Description: Technical lemma for bnj150 30200. 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 3458	. . 3 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ [𝐹 / 𝑓][1𝑜 / 𝑛]𝜑)
4		bnj125.1	. . . . . 6 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
5		bnj105 30044	. . . . . 6 ⊢ 1𝑜 ∈ V
6	4, 5	bnj91 30185	. . . . 5 ⊢ ([1𝑜 / 𝑛]𝜑 ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
7	6	sbcbii 3458	. . . 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 30188	. . . . 5 ⊢ 𝐹 ∈ V
10		fveq1 6102	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘∅) = (𝐹‘∅))
11	10	eqeq1d 2612	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅)))
12	9, 11	sbcie 3437	. . . 4 ⊢ ([𝐹 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
13	7, 12	bitri 263	. . 3 ⊢ ([𝐹 / 𝑓][1𝑜 / 𝑛]𝜑 ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
14	3, 13	bitri 263	. 2 ⊢ ([𝐹 / 𝑓]𝜑′ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))
15	1, 14	bitri 263	1 ⊢ (𝜑″ ↔ (𝐹‘∅) = pred(𝑥, 𝐴, 𝑅))

Syntax hints:   ↔ wb 195   = wceq 1475  [wsbc 3402  ∅c0 3874  {csn 4125  ⟨cop 4131  ‘cfv 5804  1_𝑜c1o 7440   predc-bnj14 30007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rex 2902  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-pw 4110  df-sn 4126  df-pr 4128  df-uni 4373  df-br 4584  df-suc 5646  df-iota 5768  df-fv 5812  df-1o 7447

This theorem is referenced by:  bnj150  30200  bnj153  30204