Theorem sbex 2463

Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)

Assertion

Ref	Expression
sbex	⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

Distinct variable groups:   𝑥,𝑦   𝑥,𝑧

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem sbex

Step	Hyp	Ref	Expression
1		sbn 2391	. . 3 ⊢ ([𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑 ↔ ¬ [𝑧 / 𝑦]∀𝑥 ¬ 𝜑)
2		sbal 2462	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥 ¬ 𝜑 ↔ ∀𝑥[𝑧 / 𝑦] ¬ 𝜑)
3		sbn 2391	. . . . 5 ⊢ ([𝑧 / 𝑦] ¬ 𝜑 ↔ ¬ [𝑧 / 𝑦]𝜑)
4	3	albii 1747	. . . 4 ⊢ (∀𝑥[𝑧 / 𝑦] ¬ 𝜑 ↔ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
5	2, 4	bitri 264	. . 3 ⊢ ([𝑧 / 𝑦]∀𝑥 ¬ 𝜑 ↔ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
6	1, 5	xchbinx 324	. 2 ⊢ ([𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
7		df-ex 1705	. . 3 ⊢ (∃𝑥𝜑 ↔ ¬ ∀𝑥 ¬ 𝜑)
8	7	sbbii 1887	. 2 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦] ¬ ∀𝑥 ¬ 𝜑)
9		df-ex 1705	. 2 ⊢ (∃𝑥[𝑧 / 𝑦]𝜑 ↔ ¬ ∀𝑥 ¬ [𝑧 / 𝑦]𝜑)
10	6, 8, 9	3bitr4i 292	1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1481  ∃wex 1704  [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbmo  2515  sbabel  2793  sbcex2  3486  sbcexgOLD  38753