Theorem bj-ax12v3 32675

Description: A weak version of ax-12 2047 which is stronger than ax12v 2048. Note that if one assumes reflexivity of equality |- x = x (equid 1939), then bj-ax12v3 32675 implies ax-5 1839 over modal logic K (substitute x for y). See also bj-ax12v3ALT 32676. (Contributed by BJ, 6-Jul-2021.)

Assertion

Ref	Expression
bj-ax12v3	|- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Distinct variable group:   ph, y

Allowed substitution hint:   ph( x)

Proof of Theorem bj-ax12v3

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 |- ( ph -> A. y ph )
2		ax12 2304	. 2 |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
3	1, 2	syl5 34	1 |- ( x = y -> ( ph -> A. x ( x = y -> ph ) ) )

Syntax hints:   -> wi 4   A.wal 1481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)