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Theorem bnj206 31346
 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 3720 . 2 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒))
2 bnj206.1 . . . 4 (𝜑′[𝑀 / 𝑛]𝜑)
32bicomi 216 . . 3 ([𝑀 / 𝑛]𝜑𝜑′)
4 bnj206.2 . . . 4 (𝜓′[𝑀 / 𝑛]𝜓)
54bicomi 216 . . 3 ([𝑀 / 𝑛]𝜓𝜓′)
6 bnj206.3 . . . 4 (𝜒′[𝑀 / 𝑛]𝜒)
76bicomi 216 . . 3 ([𝑀 / 𝑛]𝜒𝜒′)
83, 5, 73anbi123i 1200 . 2 (([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒) ↔ (𝜑′𝜓′𝜒′))
91, 8bitri 267 1 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ (𝜑′𝜓′𝜒′))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 198   ∧ w3a 1113   ∈ wcel 2166  Vcvv 3414  [wsbc 3662 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416  df-sbc 3663 This theorem is referenced by:  bnj124  31487  bnj207  31497
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