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Theorem bnj154 31465
Description: Technical lemma for bnj153 31467. 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 3689 . 2 ([𝑔 / 𝑓]𝜑′[𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4 vex 3388 . . 3 𝑔 ∈ V
5 fveq1 6410 . . . 4 (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
65eqeq1d 2801 . . 3 (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
74, 6sbcie 3668 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
81, 3, 73bitri 289 1 (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1653  [wsbc 3633  c0 4115  cfv 6101   predc-bnj14 31274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-v 3387  df-sbc 3634  df-uni 4629  df-br 4844  df-iota 6064  df-fv 6109
This theorem is referenced by:  bnj153  31467  bnj580  31500  bnj607  31503
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