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Theorem bnj206 34271
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 3842 . 2 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒))
2 bnj206.1 . . . 4 (𝜑′[𝑀 / 𝑛]𝜑)
32bicomi 223 . . 3 ([𝑀 / 𝑛]𝜑𝜑′)
4 bnj206.2 . . . 4 (𝜓′[𝑀 / 𝑛]𝜓)
54bicomi 223 . . 3 ([𝑀 / 𝑛]𝜓𝜓′)
6 bnj206.3 . . . 4 (𝜒′[𝑀 / 𝑛]𝜒)
76bicomi 223 . . 3 ([𝑀 / 𝑛]𝜒𝜒′)
83, 5, 73anbi123i 1152 . 2 (([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒) ↔ (𝜑′𝜓′𝜒′))
91, 8bitri 275 1 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))
Colors of variables: wff setvar class
Syntax hints:  wb 205  w3a 1084  wcel 2098  Vcvv 3468  [wsbc 3772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1086  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-v 3470  df-sbc 3773
This theorem is referenced by:  bnj124  34411  bnj207  34421
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