Theorem hbex 1636

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 1495	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
2	1	hbal 1477	. 2 ⊢ (∀𝑥∃𝑦𝜑 → ∀𝑦∀𝑥∃𝑦𝜑)
3		hbex.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4		19.8a 1590	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
5	3, 4	alrimih 1469	. 2 ⊢ (𝜑 → ∀𝑥∃𝑦𝜑)
6	2, 5	exlimih 1593	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1351  ∃wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  nfex  1637  excomim  1663  19.12  1665  cbvexh  1755  cbvexdh  1926  hbsbv  1941  hbeu1  2036  hbmo  2065  moexexdc  2110