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Theorem bnj156 31611
 Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj156.1 (𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))
bnj156.2 (𝜁1[𝑔 / 𝑓]𝜁0)
bnj156.3 (𝜑1[𝑔 / 𝑓]𝜑′)
bnj156.4 (𝜓1[𝑔 / 𝑓]𝜓′)
Assertion
Ref Expression
bnj156 (𝜁1 ↔ (𝑔 Fn 1o𝜑1𝜓1))

Proof of Theorem bnj156
StepHypRef Expression
1 bnj156.2 . 2 (𝜁1[𝑔 / 𝑓]𝜁0)
2 bnj156.1 . . . 4 (𝜁0 ↔ (𝑓 Fn 1o𝜑′𝜓′))
32sbcbii 3763 . . 3 ([𝑔 / 𝑓]𝜁0[𝑔 / 𝑓](𝑓 Fn 1o𝜑′𝜓′))
4 sbc3an 3772 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1o𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1o[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′))
5 bnj62 31603 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 1o𝑔 Fn 1o)
6 bnj156.3 . . . . . 6 (𝜑1[𝑔 / 𝑓]𝜑′)
76bicomi 225 . . . . 5 ([𝑔 / 𝑓]𝜑′𝜑1)
8 bnj156.4 . . . . . 6 (𝜓1[𝑔 / 𝑓]𝜓′)
98bicomi 225 . . . . 5 ([𝑔 / 𝑓]𝜓′𝜓1)
105, 7, 93anbi123i 1148 . . . 4 (([𝑔 / 𝑓]𝑓 Fn 1o[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1o𝜑1𝜓1))
114, 10bitri 276 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1o𝜑′𝜓′) ↔ (𝑔 Fn 1o𝜑1𝜓1))
123, 11bitri 276 . 2 ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1o𝜑1𝜓1))
131, 12bitri 276 1 (𝜁1 ↔ (𝑔 Fn 1o𝜑1𝜓1))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 207   ∧ w3a 1080  [wsbc 3711   Fn wfn 6227  1oc1o 7953 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-op 4485  df-br 4969  df-opab 5031  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-fun 6234  df-fn 6235 This theorem is referenced by:  bnj153  31764
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