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Theorem sbccomieg 37864
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 3654 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑𝐴 ∈ V)
2 spesbc 3727 . . 3 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑)
3 sbcex 3654 . . . 4 ([𝐴 / 𝑎]𝜑𝐴 ∈ V)
43exlimiv 2021 . . 3 (∃𝑏[𝐴 / 𝑎]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑𝐴 ∈ V)
6 nfcv 2959 . . . 4 𝑎𝐶
7 nfsbc1v 3664 . . . 4 𝑎[𝐴 / 𝑎]𝜑
86, 7nfsbc 3666 . . 3 𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑
9 sbccomieg.1 . . . 4 (𝑎 = 𝐴𝐵 = 𝐶)
10 sbceq1a 3655 . . . 4 (𝑎 = 𝐴 → (𝜑[𝐴 / 𝑎]𝜑))
119, 10sbceqbid 3651 . . 3 (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
128, 11sbciegf 3676 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
131, 5, 12pm5.21nii 369 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197   = wceq 1637  wex 1859  wcel 2157  Vcvv 3402  [wsbc 3644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-sbc 3645
This theorem is referenced by:  2rexfrabdioph  37867  3rexfrabdioph  37868  4rexfrabdioph  37869  6rexfrabdioph  37870  7rexfrabdioph  37871
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