Theorem axL4ex-P7 946

Description: Existential Form of axL4-P7 945. †

Assertion

Ref	Expression
axL4ex-P7	⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Proof of Theorem axL4ex-P7

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ndnfrall1-P7.7 832	. . . . . 6 ⊢ Ⅎ𝑥∀𝑥(𝜑 → 𝜓)
2		alle-P7.CL 942	. . . . . 6 ⊢ (∀𝑥(𝜑 → 𝜓) → (𝜑 → 𝜓))
3	1, 2	lemma-L7.02a 944	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
4		ndexi-P7.19.CL 910	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜓)
5	4	rcp-NDIMP0addall 207	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜓 → ∃𝑥𝜓))
6	3, 5	syl-P3.24 259	. . . 4 ⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜓))
7	6	rcp-NDIMP1add1 208	. . 3 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) → ([𝑦 / 𝑥]𝜑 → ∃𝑥𝜓))
8		rcp-NDASM2of2 194	. . 3 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) → ∃𝑥𝜑)
9	7, 8	ndexew-P7.VR123of3 874	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) → ∃𝑥𝜓)
10	9	rcp-NDIMI2 224	1 ⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))

Syntax hints:  term-obj 1  ∀wff-forall 8   → wff-imp 10   ∧ wff-and 132  ∃wff-exists 595  [wff-psub 714

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  ax-L10 27  ax-L11 28  ax-L12 29

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  df-psub-D6.2 716

This theorem is referenced by:  alloverimex-P7  948  qimeqallb-P7  976