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Theorem bnj934 30710
Description: Technical lemma for bnj69 30783. 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
StepHypRef Expression
1 bnj934.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 eqtr 2640 . . . 4 (((𝐺‘∅) = (𝑓‘∅) ∧ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅)) → (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
31, 2sylan2b 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 3189 . . . . . . . 8 𝑝 ∈ V
71, 5, 6bnj523 30662 . . . . . . 7 (𝜑′ ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
87, 1bitr4i 267 . . . . . 6 (𝜑′𝜑)
98sbcbii 3473 . . . . 5 ([𝐺 / 𝑓]𝜑′[𝐺 / 𝑓]𝜑)
104, 9bitri 264 . . . 4 (𝜑″[𝐺 / 𝑓]𝜑)
11 bnj934.50 . . . 4 𝐺 ∈ V
121, 10, 11bnj609 30692 . . 3 (𝜑″ ↔ (𝐺‘∅) = pred(𝑋, 𝐴, 𝑅))
133, 12sylibr 224 . 2 (((𝐺‘∅) = (𝑓‘∅) ∧ 𝜑) → 𝜑″)
1413ancoms 469 1 ((𝜑 ∧ (𝐺‘∅) = (𝑓‘∅)) → 𝜑″)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  [wsbc 3417  c0 3891  cfv 5847   predc-bnj14 30458
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-v 3188  df-sbc 3418  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855
This theorem is referenced by:  bnj929  30711
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