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Theorem bnj156 30050
 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 3458 . . 3 ([𝑔 / 𝑓]𝜁0[𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′))
4 sbc3an 3461 . . . 4 ([𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ ([𝑔 / 𝑓]𝑓 Fn 1𝑜[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′))
5 bnj62 30040 . . . . 5 ([𝑔 / 𝑓]𝑓 Fn 1𝑜𝑔 Fn 1𝑜)
6 bnj156.3 . . . . . 6 (𝜑1[𝑔 / 𝑓]𝜑′)
76bicomi 213 . . . . 5 ([𝑔 / 𝑓]𝜑′𝜑1)
8 bnj156.4 . . . . . 6 (𝜓1[𝑔 / 𝑓]𝜓′)
98bicomi 213 . . . . 5 ([𝑔 / 𝑓]𝜓′𝜓1)
105, 7, 93anbi123i 1244 . . . 4 (([𝑔 / 𝑓]𝑓 Fn 1𝑜[𝑔 / 𝑓]𝜑′[𝑔 / 𝑓]𝜓′) ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
114, 10bitri 263 . . 3 ([𝑔 / 𝑓](𝑓 Fn 1𝑜𝜑′𝜓′) ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
123, 11bitri 263 . 2 ([𝑔 / 𝑓]𝜁0 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
131, 12bitri 263 1 (𝜁1 ↔ (𝑔 Fn 1𝑜𝜑1𝜓1))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ w3a 1031  [wsbc 3402   Fn wfn 5799  1𝑜c1o 7440 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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-fun 5806  df-fn 5807 This theorem is referenced by:  bnj153  30204
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