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Theorem sbccom2lem 35440
 Description: Lemma for sbccom2 35441. (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 3797 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑))
2 sbc5 3777 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 sbccom2lem.1 . . . . . 6 𝐴 ∈ V
43csbconstgi 3878 . . . . . 6 𝐴 / 𝑥𝑦 = 𝑦
5 eqid 2821 . . . . . 6 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
63, 4, 5sbceqi 4335 . . . . 5 ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)
76anbi1i 626 . . . 4 (([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
81, 2, 73bitr3i 304 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
98exbii 1849 . 2 (∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
10 sbc5 3777 . . . . 5 ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵𝜑))
1110sbcbii 3805 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑))
12 sbc5 3777 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1311, 12bitri 278 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
14 19.42v 1955 . . . . . 6 (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1514bicomi 227 . . . . 5 ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1615exbii 1849 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
17 excom 2170 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1816, 17bitri 278 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1913, 18bitri 278 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
20 sbc5 3777 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
219, 19, 203bitr4i 306 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538  ∃wex 1781   ∈ wcel 2115  Vcvv 3471  [wsbc 3749  ⦋csb 3857 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-sbc 3750  df-csb 3858 This theorem is referenced by:  sbccom2  35441
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