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Theorem sbccomieg 36375
 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 3412 . 2 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑𝐴 ∈ V)
2 spesbc 3487 . . 3 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑 → ∃𝑏[𝐴 / 𝑎]𝜑)
3 sbcex 3412 . . . 4 ([𝐴 / 𝑎]𝜑𝐴 ∈ V)
43exlimiv 1845 . . 3 (∃𝑏[𝐴 / 𝑎]𝜑𝐴 ∈ V)
52, 4syl 17 . 2 ([𝐶 / 𝑏][𝐴 / 𝑎]𝜑𝐴 ∈ V)
6 nfcv 2751 . . . 4 𝑎𝐶
7 nfsbc1v 3422 . . . 4 𝑎[𝐴 / 𝑎]𝜑
86, 7nfsbc 3424 . . 3 𝑎[𝐶 / 𝑏][𝐴 / 𝑎]𝜑
9 sbccomieg.1 . . . 4 (𝑎 = 𝐴𝐵 = 𝐶)
10 sbceq1a 3413 . . . 4 (𝑎 = 𝐴 → (𝜑[𝐴 / 𝑎]𝜑))
119, 10sbceqbid 3409 . . 3 (𝑎 = 𝐴 → ([𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
128, 11sbciegf 3434 . 2 (𝐴 ∈ V → ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑))
131, 5, 12pm5.21nii 367 1 ([𝐴 / 𝑎][𝐵 / 𝑏]𝜑[𝐶 / 𝑏][𝐴 / 𝑎]𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-sbc 3403 This theorem is referenced by:  2rexfrabdioph  36378  3rexfrabdioph  36379  4rexfrabdioph  36380  6rexfrabdioph  36381  7rexfrabdioph  36382
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