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Theorem sbccomieg 36876
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 3432 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑𝐴 ∈ V)
2 spesbc 3507 . . 3 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑)
3 sbcex 3432 . . . 4 ([𝐴 / 𝑎]𝜑𝐴 ∈ V)
43exlimiv 1855 . . 3 (∃𝑏[𝐴 / 𝑎]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑𝐴 ∈ V)
6 nfcv 2761 . . . 4 𝑎𝐶
7 nfsbc1v 3442 . . . 4 𝑎[𝐴 / 𝑎]𝜑
86, 7nfsbc 3444 . . 3 𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑
9 sbccomieg.1 . . . 4 (𝑎 = 𝐴𝐵 = 𝐶)
10 sbceq1a 3433 . . . 4 (𝑎 = 𝐴 → (𝜑[𝐴 / 𝑎]𝜑))
119, 10sbceqbid 3429 . . 3 (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
128, 11sbciegf 3454 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
131, 5, 12pm5.21nii 368 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196   = wceq 1480  wex 1701  wcel 1987  Vcvv 3190  [wsbc 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423
This theorem is referenced by:  2rexfrabdioph  36879  3rexfrabdioph  36880  4rexfrabdioph  36881  6rexfrabdioph  36882  7rexfrabdioph  36883
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