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Theorem sbccom2lem 32895
Description: Lemma for sbccom2 32896. (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 3444 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑))
2 sbc5 3426 . . . 4 ([𝐴 / 𝑥](𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
3 sbccom2lem.1 . . . . . 6 𝐴 ∈ V
43csbconstgi 32888 . . . . . 6 𝐴 / 𝑥𝑦 = 𝑦
5 eqid 2609 . . . . . 6 𝐴 / 𝑥𝐵 = 𝐴 / 𝑥𝐵
63, 4, 5sbceqi 32879 . . . . 5 ([𝐴 / 𝑥]𝑦 = 𝐵𝑦 = 𝐴 / 𝑥𝐵)
76anbi1i 726 . . . 4 (([𝐴 / 𝑥]𝑦 = 𝐵[𝐴 / 𝑥]𝜑) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
81, 2, 73bitr3i 288 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
98exbii 1763 . 2 (∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
10 sbc5 3426 . . . . 5 ([𝐵 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝐵𝜑))
1110sbcbii 3457 . . . 4 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑))
12 sbc5 3426 . . . 4 ([𝐴 / 𝑥]𝑦(𝑦 = 𝐵𝜑) ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1311, 12bitri 262 . . 3 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
14 19.42v 1904 . . . . . 6 (∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ (𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)))
1514bicomi 212 . . . . 5 ((𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1615exbii 1763 . . . 4 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
17 excom 2028 . . . 4 (∃𝑥𝑦(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1816, 17bitri 262 . . 3 (∃𝑥(𝑥 = 𝐴 ∧ ∃𝑦(𝑦 = 𝐵𝜑)) ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
1913, 18bitri 262 . 2 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑 ↔ ∃𝑦𝑥(𝑥 = 𝐴 ∧ (𝑦 = 𝐵𝜑)))
20 sbc5 3426 . 2 ([𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑 ↔ ∃𝑦(𝑦 = 𝐴 / 𝑥𝐵[𝐴 / 𝑥]𝜑))
219, 19, 203bitr4i 290 1 ([𝐴 / 𝑥][𝐵 / 𝑦]𝜑[𝐴 / 𝑥𝐵 / 𝑦][𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  Vcvv 3172  [wsbc 3401  csb 3498
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-sbc 3402  df-csb 3499
This theorem is referenced by:  sbccom2  32896
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