Theorem qremexw-P6 703

Description: Existential Quantifier Removal (non-freeness condition - weakened form).

Requires the existence of '𝜑₁(𝑥₁)' as a replacement for '𝜑(𝑥)'.

Hypotheses

Ref	Expression
qremexw-P6.1	⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
qremexw-P6.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
qremexw-P6	⊢ (∃𝑥𝜑 ↔ 𝜑)

Distinct variable groups:   𝜑,𝑥₁   𝜑₁,𝑥   𝑥,𝑥₁

Proof of Theorem qremexw-P6

Step	Hyp	Ref	Expression
1		qremexw-P6.1	. . 3 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2		qremexw-P6.2	. . 3 ⊢ Ⅎ𝑥𝜑
3	1, 2	nfrexgenw-P6 696	. 2 ⊢ (∃𝑥𝜑 → 𝜑)
4	1	exiw-P5 662	. 2 ⊢ (𝜑 → ∃𝑥𝜑)
5	3, 4	rcp-NDBII0 239	1 ⊢ (∃𝑥𝜑 ↔ 𝜑)

Syntax hints:  term-obj 1   = wff-equals 6   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  Ⅎwff-nfree 681

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  ax-L7 19

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  df-nfree-D6.1 682

This theorem is referenced by: (None)