Theorem hbe1a 2042

Description: Dual statement of hbe1 1509. (Contributed by Wolf Lammen, 15-Sep-2021.)

Assertion

Ref	Expression
hbe1a	|- ( E. x A. x ph -> A. x ph )

Proof of Theorem hbe1a

Step	Hyp	Ref	Expression
1		nfa1 1555	. . 3 |- F/ x A. x ph
2		nf3 1683	. . 3 |- ( F/ x A. x ph <-> A. x ( E. x A. x ph -> A. x ph ) )
3	1, 2	mpbi 145	. 2 |- A. x ( E. x A. x ph -> A. x ph )
4	3	spi 1550	1 |- ( E. x A. x ph -> A. x ph )

Syntax hints:   -> wi 4   A.wal 1362   F/wnf 1474   E.wex 1506

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 1461  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-4 1524  ax-ial 1548  ax-i5r 1549

This theorem depends on definitions:  df-bi 117  df-nf 1475

This theorem is referenced by:  nf5-1  2043