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Theorem sbccomieg 42781
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 3801 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑𝐴 ∈ V)
2 spesbc 3891 . . 3 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑)
3 sbcex 3801 . . . 4 ([𝐴 / 𝑎]𝜑𝐴 ∈ V)
43exlimiv 1928 . . 3 (∃𝑏[𝐴 / 𝑎]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑𝐴 ∈ V)
6 nfcv 2903 . . . 4 𝑎𝐶
7 nfsbc1v 3811 . . . 4 𝑎[𝐴 / 𝑎]𝜑
86, 7nfsbcw 3813 . . 3 𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑
9 sbccomieg.1 . . . 4 (𝑎 = 𝐴𝐵 = 𝐶)
10 sbceq1a 3802 . . . 4 (𝑎 = 𝐴 → (𝜑[𝐴 / 𝑎]𝜑))
119, 10sbceqbid 3798 . . 3 (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
128, 11sbciegf 3831 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
131, 5, 12pm5.21nii 378 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wex 1776  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-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-v 3480  df-sbc 3792
This theorem is referenced by:  2rexfrabdioph  42784  3rexfrabdioph  42785  4rexfrabdioph  42786  6rexfrabdioph  42787  7rexfrabdioph  42788
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