Theorem alloverimex-P5 601

Description: Alternate version of alloverim-P5 588 that produces existential quantifiers.

Hypothesis

Ref	Expression
alloverimex-P5.1	⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))

Assertion

Ref	Expression
alloverimex-P5	⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem alloverimex-P5

Step	Hyp	Ref	Expression
1		alloverimex-P5.1	. . . . 5 ⊢ (𝛾 → ∀𝑥(𝜑 → 𝜓))
2		trnspeq-P4c 537	. . . . . . 7 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜓 → ¬ 𝜑))
3	2	suballinf-P5 594	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(¬ 𝜓 → ¬ 𝜑))
4	3	rcp-NDIMP0addall 207	. . . . 5 ⊢ (𝛾 → (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑥(¬ 𝜓 → ¬ 𝜑)))
5	1, 4	bimpf-P4 531	. . . 4 ⊢ (𝛾 → ∀𝑥(¬ 𝜓 → ¬ 𝜑))
6	5	alloverim-P5 588	. . 3 ⊢ (𝛾 → (∀𝑥 ¬ 𝜓 → ∀𝑥 ¬ 𝜑))
7		allnegex-P5 597	. . . 4 ⊢ (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓)
8	7	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (∀𝑥 ¬ 𝜓 ↔ ¬ ∃𝑥𝜓))
9		allnegex-P5 597	. . . 4 ⊢ (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑)
10	9	rcp-NDIMP0addall 207	. . 3 ⊢ (𝛾 → (∀𝑥 ¬ 𝜑 ↔ ¬ ∃𝑥𝜑))
11	6, 8, 10	subimd2-P4 544	. 2 ⊢ (𝛾 → (¬ ∃𝑥𝜓 → ¬ ∃𝑥𝜑))
12	11	trnsp-P3.31d 288	1 ⊢ (𝛾 → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:  ∀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

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:  alloverimex-P5.RC  602  alloverimex-P5.CL  604  dalloverimex-P5  605  alloverimex-P5.GENV  622  alloverimex-P5.GENF  748