Theorem qremexv-P5 657

Description: Existential Quantifier Removal (variable restriction).

'𝑥' cannot occur in '𝜑'.

The most general form is qremex-P6 723.

Assertion

Ref	Expression
qremexv-P5	⊢ (∃𝑥𝜑 ↔ 𝜑)

Distinct variable group:   𝜑,𝑥

Proof of Theorem qremexv-P5

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		axL5ex-P5 613	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
2		biref-P3.33a 297	. . . 4 ⊢ (𝜑 ↔ 𝜑)
3	2	rcp-NDIMP0addall 207	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜑))
4	3	exiisub-P5 655	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	1, 4	rcp-NDBII0 239	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:  term-obj 1   = wff-equals 6   ↔ wff-bi 104  ∃wff-exists 595

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596

This theorem is referenced by:  qcallimrv-P5  671  qcallimlv-P5  673