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Theorem bnj206 32026
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 3823 . 2 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒))
2 bnj206.1 . . . 4 (𝜑′[𝑀 / 𝑛]𝜑)
32bicomi 227 . . 3 ([𝑀 / 𝑛]𝜑𝜑′)
4 bnj206.2 . . . 4 (𝜓′[𝑀 / 𝑛]𝜓)
54bicomi 227 . . 3 ([𝑀 / 𝑛]𝜓𝜓′)
6 bnj206.3 . . . 4 (𝜒′[𝑀 / 𝑛]𝜒)
76bicomi 227 . . 3 ([𝑀 / 𝑛]𝜒𝜒′)
83, 5, 73anbi123i 1152 . 2 (([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒) ↔ (𝜑′𝜓′𝜒′))
91, 8bitri 278 1 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))
Colors of variables: wff setvar class
Syntax hints:  wb 209  w3a 1084  wcel 2115  Vcvv 3480  [wsbc 3758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-ext 2796
This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-sbc 3759
This theorem is referenced by:  bnj124  32168  bnj207  32178
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