Theorem genallw-P6 676

Description: The WFF ∀𝑥𝜑 is General for '𝑥' (weakened form).

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

Hypothesis

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

Assertion

Ref	Expression
genallw-P6	⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

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

Proof of Theorem genallw-P6

Step	Hyp	Ref	Expression
1		ax-L5 17	. 2 ⊢ (∀𝑥₁𝜑₁ → ∀𝑥∀𝑥₁𝜑₁)
2		genallw-P6.1	. . . 4 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
3	2	cbvallv-P5 659	. . 3 ⊢ (∀𝑥𝜑 ↔ ∀𝑥₁𝜑₁)
4	3	bisym-P3.33b.RC 299	. 2 ⊢ (∀𝑥₁𝜑₁ ↔ ∀𝑥𝜑)
5	4	suballinf-P5 594	. 2 ⊢ (∀𝑥∀𝑥₁𝜑₁ ↔ ∀𝑥∀𝑥𝜑)
6	1, 4, 5	subimd2-P4.RC 545	1 ⊢ (∀𝑥𝜑 → ∀𝑥∀𝑥𝜑)

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

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

This theorem is referenced by:  gennexw-P6  679  nfrall1w-P6  692