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Theorem bnj154 32049
Description: Technical lemma for bnj153 32051. 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
bnj154.1 (𝜑1[𝑔 / 𝑓]𝜑′)
bnj154.2 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj154 (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓   𝑅,𝑓   𝑓,𝑔   𝑥,𝑓
Allowed substitution hints:   𝐴(𝑥,𝑔)   𝑅(𝑥,𝑔)   𝜑′(𝑥,𝑓,𝑔)   𝜑1(𝑥,𝑓,𝑔)

Proof of Theorem bnj154
StepHypRef Expression
1 bnj154.1 . 2 (𝜑1[𝑔 / 𝑓]𝜑′)
2 bnj154.2 . . 3 (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
32sbcbii 3826 . 2 ([𝑔 / 𝑓]𝜑′[𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4 vex 3495 . . 3 𝑔 ∈ V
5 fveq1 6662 . . . 4 (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
65eqeq1d 2820 . . 3 (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
74, 6sbcie 3809 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
81, 3, 73bitri 298 1 (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 207   = wceq 1528  [wsbc 3769  c0 4288  cfv 6348   predc-bnj14 31857
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-rex 3141  df-v 3494  df-sbc 3770  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356
This theorem is referenced by:  bnj153  32051  bnj580  32084  bnj607  32087
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