Theorem exgenallw-P6 680

Description: Dual of gennallw-P6 678 (weakened form).

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

Hypothesis

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

Assertion

Ref	Expression
exgenallw-P6	⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

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

Proof of Theorem exgenallw-P6

Step	Hyp	Ref	Expression
1		exgenallw-P6.1	. . 3 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑₁))
2	1	gennallw-P6 678	. 2 ⊢ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)
3		lemma-L5.01a 600	. 2 ⊢ ((∃𝑥∀𝑥𝜑 → ∀𝑥𝜑) ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑))
4	2, 3	bimpr-P4.RC 534	1 ⊢ (∃𝑥∀𝑥𝜑 → ∀𝑥𝜑)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8  ¬ wff-neg 9   → wff-imp 10   ↔ 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  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:  gennfrw-P6  685