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Theorem sbccom2lem 33099
 Description: Lemma for sbccom2 33100. (Contributed by Giovanni Mascellani, 31-May-2019.)
Hypothesis
Ref Expression
sbccom2lem.1 𝐴 ∈ V
Assertion
Ref Expression
sbccom2lem ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑦,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐵(𝑥)

Proof of Theorem sbccom2lem
StepHypRef Expression
1 sbcan 3445 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑))
2 sbc5 3427 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 sbccom2lem.1 . . . . . 6 𝐴 ∈ V
43csbconstgi 33092 . . . . . 6 𝐴 / 𝑥𝑦 = 𝑦
5 eqid 2610 . . . . . 6 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
63, 4, 5sbceqi 33083 . . . . 5 ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)
76anbi1i 727 . . . 4 (([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
81, 2, 73bitr3i 289 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
98exbii 1764 . 2 (∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
10 sbc5 3427 . . . . 5 ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵𝜑))
1110sbcbii 3458 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑))
12 sbc5 3427 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1311, 12bitri 263 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
14 19.42v 1905 . . . . . 6 (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1514bicomi 213 . . . . 5 ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1615exbii 1764 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
17 excom 2029 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1816, 17bitri 263 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1913, 18bitri 263 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
20 sbc5 3427 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
219, 19, 203bitr4i 291 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  [wsbc 3402  ⦋csb 3499 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-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403  df-csb 3500 This theorem is referenced by:  sbccom2  33100
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