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Theorem sbccomieg 41019
Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
sbccomieg.1 (𝑎 = 𝐴𝐵 = 𝐶)
Assertion
Ref Expression
sbccomieg ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)
Distinct variable groups:   𝐴,𝑎,𝑏   𝐶,𝑎
Allowed substitution hints:   𝜑(𝑎,𝑏)   𝐵(𝑎,𝑏)   𝐶(𝑏)

Proof of Theorem sbccomieg
StepHypRef Expression
1 sbcex 3748 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑𝐴 ∈ V)
2 spesbc 3837 . . 3 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑)
3 sbcex 3748 . . . 4 ([𝐴 / 𝑎]𝜑𝐴 ∈ V)
43exlimiv 1934 . . 3 (∃𝑏[𝐴 / 𝑎]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑𝐴 ∈ V)
6 nfcv 2906 . . . 4 𝑎𝐶
7 nfsbc1v 3758 . . . 4 𝑎[𝐴 / 𝑎]𝜑
86, 7nfsbcw 3760 . . 3 𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑
9 sbccomieg.1 . . . 4 (𝑎 = 𝐴𝐵 = 𝐶)
10 sbceq1a 3749 . . . 4 (𝑎 = 𝐴 → (𝜑[𝐴 / 𝑎]𝜑))
119, 10sbceqbid 3745 . . 3 (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
128, 11sbciegf 3777 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
131, 5, 12pm5.21nii 380 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wex 1782  wcel 2107  Vcvv 3444  [wsbc 3738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ral 3064  df-rex 3073  df-v 3446  df-sbc 3739
This theorem is referenced by:  2rexfrabdioph  41022  3rexfrabdioph  41023  4rexfrabdioph  41024  6rexfrabdioph  41025  7rexfrabdioph  41026
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