Theorem hbex 1568

Description: If 𝑥 is not free in 𝜑, it is not free in ∃𝑦𝜑. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 2-Feb-2015.)

Hypothesis

Ref	Expression
hbex.1	⊢ (𝜑 → ∀𝑥𝜑)

Assertion

Ref	Expression
hbex	⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Proof of Theorem hbex

Step	Hyp	Ref	Expression
1		hbe1 1425	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
2	1	hbal 1407	. 2 ⊢ (∀𝑥∃𝑦𝜑 → ∀𝑦∀𝑥∃𝑦𝜑)
3		hbex.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4		19.8a 1523	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
5	3, 4	alrimih 1399	. 2 ⊢ (𝜑 → ∀𝑥∃𝑦𝜑)
6	2, 5	exlimih 1525	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1283  ∃wex 1422

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441

This theorem depends on definitions:  df-bi 115

This theorem is referenced by:  nfex  1569  excomim  1594  19.12  1596  cbvexh  1679  cbvexdh  1843  hbsbv  1859  hbeu1  1952  hbmo  1981  moexexdc  2026