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Theorem bnj154 31286
Description: Technical lemma for bnj153 31288. 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 3643 . 2 ([𝑔 / 𝑓]𝜑′[𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
4 vex 3354 . . 3 𝑔 ∈ V
5 fveq1 6331 . . . 4 (𝑓 = 𝑔 → (𝑓‘∅) = (𝑔‘∅))
65eqeq1d 2773 . . 3 (𝑓 = 𝑔 → ((𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅)))
74, 6sbcie 3622 . 2 ([𝑔 / 𝑓](𝑓‘∅) = pred(𝑥, 𝐴, 𝑅) ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
81, 3, 73bitri 286 1 (𝜑1 ↔ (𝑔‘∅) = pred(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1631  [wsbc 3587  c0 4063  cfv 6031   predc-bnj14 31094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-v 3353  df-sbc 3588  df-uni 4575  df-br 4787  df-iota 5994  df-fv 6039
This theorem is referenced by:  bnj153  31288  bnj580  31321  bnj607  31324
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