Theorem sbex 1979

Description: Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.) (Proof rewritten by Jim Kingdon, 12-Feb-2018.)

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

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbexyz 1978	. . . 4 ⊢ ([𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑤 / 𝑦]𝜑)
2	1	sbbii 1738	. . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑)
3		sbexyz 1978	. . 3 ⊢ ([𝑧 / 𝑤]∃𝑥[𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
4	2, 3	bitri 183	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑)
5		ax-17 1506	. . 3 ⊢ (∃𝑥𝜑 → ∀𝑤∃𝑥𝜑)
6	5	sbco2vh 1918	. 2 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]∃𝑥𝜑 ↔ [𝑧 / 𝑦]∃𝑥𝜑)
7		ax-17 1506	. . . 4 ⊢ (𝜑 → ∀𝑤𝜑)
8	7	sbco2vh 1918	. . 3 ⊢ ([𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ [𝑧 / 𝑦]𝜑)
9	8	exbii 1584	. 2 ⊢ (∃𝑥[𝑧 / 𝑤][𝑤 / 𝑦]𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)
10	4, 6, 9	3bitr3i 209	1 ⊢ ([𝑧 / 𝑦]∃𝑥𝜑 ↔ ∃𝑥[𝑧 / 𝑦]𝜑)

Syntax hints:   ↔ wb 104  ∃wex 1468  [wsb 1735

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736

This theorem is referenced by:  sbabel  2307  sbcex2  2962  sbcexg  2963