Theorem qremallw-P6 702

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

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

Hypotheses

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

Assertion

Ref	Expression
qremallw-P6	⊢ (∀𝑥𝜑 ↔ 𝜑)

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

Proof of Theorem qremallw-P6

Step	Hyp	Ref	Expression
1		qremallw-P6.1	. . 3 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	specw-P5 661	. 2 ⊢ (∀𝑥𝜑 → 𝜑)
3		qremallw-P6.2	. . 3 ⊢ Ⅎ𝑥𝜑
4	1, 3	nfrgenw-P6 684	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
5	2, 4	rcp-NDBII0 239	1 ⊢ (∀𝑥𝜑 ↔ 𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  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:  solvesub-P6a  704