Theorem drnf1 1969

Description: Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
dral1.1	|- (A.x x = y -> (ph <-> ps))

Assertion

Ref	Expression
drnf1	|- (A.x x = y -> ( F/xph <-> F/yps))

Proof of Theorem drnf1

Step	Hyp	Ref	Expression
1		dral1.1	. . . 4 |- (A.x x = y -> (ph <-> ps))
2	1	dral1 1965	. . . 4 |- (A.x x = y -> (A.xph <-> A.yps))
3	1, 2	imbi12d 311	. . 3 |- (A.x x = y -> ((ph -> A.xph) <-> (ps -> A.yps)))
4	3	dral1 1965	. 2 |- (A.x x = y -> (A.x(ph -> A.xph) <-> A.y(ps -> A.yps)))
5		df-nf 1545	. 2 |- ( F/xph <-> A.x(ph -> A.xph))
6		df-nf 1545	. 2 |- ( F/yps <-> A.y(ps -> A.yps))
7	4, 5, 6	3bitr4g 279	1 |- (A.x x = y -> ( F/xph <-> F/yps))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   F/wnf 1544

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  nfald2  1972  drnfc1  2505