Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj206 Structured version   Visualization version   GIF version

Theorem bnj206 34724
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 3861 . 2 ([𝑀 / 𝑛](𝜑𝜓𝜒) ↔ ([𝑀 / 𝑛]𝜑[𝑀 / 𝑛]𝜓[𝑀 / 𝑛]𝜒))
2 bnj206.1 . . . 4 (𝜑′[𝑀 / 𝑛]𝜑)
32bicomi 224 . . 3 ([𝑀 / 𝑛]𝜑𝜑′)
4 bnj206.2 . . . 4 (𝜓′[𝑀 / 𝑛]𝜓)
54bicomi 224 . . 3 ([𝑀 / 𝑛]𝜓𝜓′)
6 bnj206.3 . . . 4 (𝜒′[𝑀 / 𝑛]𝜒)
76bicomi 224 . . 3 ([𝑀 / 𝑛]𝜒𝜒′)
83, 5, 73anbi123i 1154 . 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 206  w3a 1086  wcel 2106  Vcvv 3478  [wsbc 3791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-sbc 3792
This theorem is referenced by:  bnj124  34864  bnj207  34874
  Copyright terms: Public domain W3C validator