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Theorem bnj206 30053
Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj206.1 (𝜑′[𝑀 / 𝑛]𝜑)
bnj206.2 (𝜓′[𝑀 / 𝑛]𝜓)
bnj206.3 (𝜒′[𝑀 / 𝑛]𝜒)
bnj206.4 𝑀 ∈ V
Assertion
Ref Expression
bnj206 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))

Proof of Theorem bnj206
StepHypRef Expression
1 sbc3an 3461 . 2 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒))
2 bnj206.1 . . . 4 (𝜑′[𝑀 / 𝑛]𝜑)
32bicomi 213 . . 3 ([𝑀 / 𝑛]𝜑𝜑′)
4 bnj206.2 . . . 4 (𝜓′[𝑀 / 𝑛]𝜓)
54bicomi 213 . . 3 ([𝑀 / 𝑛]𝜓𝜓′)
6 bnj206.3 . . . 4 (𝜒′[𝑀 / 𝑛]𝜒)
76bicomi 213 . . 3 ([𝑀 / 𝑛]𝜒𝜒′)
83, 5, 73anbi123i 1244 . 2 (([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒) ↔ (𝜑′𝜓′𝜒′))
91, 8bitri 263 1 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))
Colors of variables: wff setvar class
Syntax hints:  wb 195  w3a 1031  wcel 1977  Vcvv 3173  [wsbc 3402
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-v 3175  df-sbc 3403
This theorem is referenced by:  bnj124  30195  bnj207  30205
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