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Theorem 2sb5 2442
Description: Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
Assertion
Ref Expression
2sb5 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑦,𝑤
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem 2sb5
StepHypRef Expression
1 sb5 2429 . 2 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥(𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑))
2 19.42v 1915 . . . 4 (∃𝑦(𝑥 = 𝑧 ∧ (𝑦 = 𝑤𝜑)) ↔ (𝑥 = 𝑧 ∧ ∃𝑦(𝑦 = 𝑤𝜑)))
3 anass 680 . . . . 5 (((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑) ↔ (𝑥 = 𝑧 ∧ (𝑦 = 𝑤𝜑)))
43exbii 1771 . . . 4 (∃𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑) ↔ ∃𝑦(𝑥 = 𝑧 ∧ (𝑦 = 𝑤𝜑)))
5 sb5 2429 . . . . 5 ([𝑤 / 𝑦]𝜑 ↔ ∃𝑦(𝑦 = 𝑤𝜑))
65anbi2i 729 . . . 4 ((𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ (𝑥 = 𝑧 ∧ ∃𝑦(𝑦 = 𝑤𝜑)))
72, 4, 63bitr4ri 293 . . 3 ((𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ ∃𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
87exbii 1771 . 2 (∃𝑥(𝑥 = 𝑧 ∧ [𝑤 / 𝑦]𝜑) ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
91, 8bitri 264 1 ([𝑧 / 𝑥][𝑤 / 𝑦]𝜑 ↔ ∃𝑥𝑦((𝑥 = 𝑧𝑦 = 𝑤) ∧ 𝜑))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wex 1701  [wsb 1877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-sb 1878
This theorem is referenced by:  opelopabsbALT  4949
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