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Theorem sbccom2lem 34059
Description: Lemma for sbccom2 34060. (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 3511 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑))
2 sbc5 3493 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 sbccom2lem.1 . . . . . 6 𝐴 ∈ V
43csbconstgi 34052 . . . . . 6 𝐴 / 𝑥𝑦 = 𝑦
5 eqid 2651 . . . . . 6 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
63, 4, 5sbceqi 34043 . . . . 5 ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)
76anbi1i 731 . . . 4 (([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
81, 2, 73bitr3i 290 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
98exbii 1814 . 2 (∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
10 sbc5 3493 . . . . 5 ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵𝜑))
1110sbcbii 3524 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑))
12 sbc5 3493 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1311, 12bitri 264 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
14 19.42v 1921 . . . . . 6 (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1514bicomi 214 . . . . 5 ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1615exbii 1814 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
17 excom 2082 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1816, 17bitri 264 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1913, 18bitri 264 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
20 sbc5 3493 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
219, 19, 203bitr4i 292 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  Vcvv 3231  [wsbc 3468  csb 3566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567
This theorem is referenced by:  sbccom2  34060
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