Theorem hbex 1615

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 1471	. . 3 ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑)
2	1	hbal 1453	. 2 ⊢ (∀𝑥∃𝑦𝜑 → ∀𝑦∀𝑥∃𝑦𝜑)
3		hbex.1	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
4		19.8a 1569	. . 3 ⊢ (𝜑 → ∃𝑦𝜑)
5	3, 4	alrimih 1445	. 2 ⊢ (𝜑 → ∀𝑥∃𝑦𝜑)
6	2, 5	exlimih 1572	1 ⊢ (∃𝑦𝜑 → ∀𝑥∃𝑦𝜑)

Syntax hints:   → wi 4  ∀wal 1329  ∃wex 1468

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-4 1487

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  nfex  1616  excomim  1641  19.12  1643  cbvexh  1728  cbvexdh  1898  hbsbv  1914  hbeu1  2009  hbmo  2038  moexexdc  2083