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Theorem bnj154 30202
 Description: Technical lemma for bnj153 30204. 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 3458 . 2 ([𝑔 / 𝑓]𝜑′[𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4 vex 3176 . . 3 𝑔 ∈ V
5 fveq1 6102 . . . 4 (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
65eqeq1d 2612 . . 3 (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
74, 6sbcie 3437 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
81, 3, 73bitri 285 1 (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  [wsbc 3402  ∅c0 3874  ‘cfv 5804   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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 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-uni 4373  df-br 4584  df-iota 5768  df-fv 5812 This theorem is referenced by:  bnj153  30204  bnj580  30237  bnj607  30240
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