Theorem bnj934 31005

Description: Technical lemma for bnj69 31078. 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
bnj934.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj934.4	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj934.7	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj934.50	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
bnj934	⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)

Distinct variable groups:   𝐴,𝑓,𝑛   𝑅,𝑓,𝑛   𝑓,𝑋,𝑛

Allowed substitution hints:   𝜑(𝑓,𝑛,𝑝)   𝐴(𝑝)   𝑅(𝑝)   𝐺(𝑓,𝑛,𝑝)   𝑋(𝑝)   𝜑′(𝑓,𝑛,𝑝)   𝜑″(𝑓,𝑛,𝑝)

Proof of Theorem bnj934

Step	Hyp	Ref	Expression
1		bnj934.1	. . . 4 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		eqtr 2641	. . . 4 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
3	1, 2	sylan2b 492	. . 3 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
4		bnj934.7	. . . . 5 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
5		bnj934.4	. . . . . . . 8 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
6		vex 3203	. . . . . . . 8 ⊢ 𝑝 ∈ V
7	1, 5, 6	bnj523 30957	. . . . . . 7 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
8	7, 1	bitr4i 267	. . . . . 6 ⊢ (𝜑′ ↔ 𝜑)
9	8	sbcbii 3491	. . . . 5 ⊢ ([𝐺 / 𝑓]𝜑′ ↔ [𝐺 / 𝑓]𝜑)
10	4, 9	bitri 264	. . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑)
11		bnj934.50	. . . 4 ⊢ 𝐺 ∈ V
12	1, 10, 11	bnj609 30987	. . 3 ⊢ (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
13	3, 12	sylibr 224	. 2 ⊢ (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → 𝜑″)
14	13	ancoms 469	1 ⊢ ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  [wsbc 3435  ∅c0 3915  ‘cfv 5888   predc-bnj14 30754

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-v 3202  df-sbc 3436  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896

This theorem is referenced by:  bnj929  31006