Theorem 19.9d2r 29319

Description: A deduction version of one direction of 19.9 2072 with two variables. (Contributed by Thierry Arnoux, 30-Jan-2017.)

Hypotheses

Ref	Expression
19.9d2r.1	|- ( ph -> F/ x ps )
19.9d2r.2	|- ( ph -> F/ y ps )
19.9d2r.3	|- ( ph -> E. x e. A E. y e. B ps )

Assertion

Ref	Expression
19.9d2r	|- ( ph -> ps )

Distinct variable group:   ph, y

Allowed substitution hints:   ph( x)   ps( x, y)   A( x, y)   B( x, y)

Proof of Theorem 19.9d2r

Step	Hyp	Ref	Expression
1		nfv 1843	. 2 |- F/ y ph
2		19.9d2r.1	. 2 |- ( ph -> F/ x ps )
3		19.9d2r.2	. 2 |- ( ph -> F/ y ps )
4		19.9d2r.3	. 2 |- ( ph -> E. x e. A E. y e. B ps )
5	1, 2, 3, 4	19.9d2rf 29318	1 |- ( ph -> ps )

Syntax hints:   -> wi 4   F/wnf 1708   E.wrex 2913

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-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-rex 2918

This theorem is referenced by:  xrofsup  29533