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Theorem bnj154 32235
 Description: Technical lemma for bnj153 32237. 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 3814 . 2 ([𝑔 / 𝑓]𝜑′[𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4 vex 3483 . . 3 𝑔 ∈ V
5 fveq1 6662 . . . 4 (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
65eqeq1d 2826 . . 3 (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
74, 6sbcie 3798 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
81, 3, 73bitri 300 1 (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  [wsbc 3758  ∅c0 4276  ‘cfv 6345   predc-bnj14 32043 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759  df-in 3926  df-ss 3936  df-uni 4825  df-br 5054  df-iota 6304  df-fv 6353 This theorem is referenced by:  bnj153  32237  bnj580  32270  bnj607  32273
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