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Theorem equsalhwOLD 1839
 Description: Obsolete proof of equsalhw 1838 as of 28-Dec-2017. (Contributed by NM, 29-Nov-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
equsalhwOLD.1 (ψxψ)
equsalhwOLD.2 (x = y → (φψ))
Assertion
Ref Expression
equsalhwOLD (x(x = yφ) ↔ ψ)
Distinct variable group:   x,y
Allowed substitution hints:   φ(x,y)   ψ(x,y)

Proof of Theorem equsalhwOLD
StepHypRef Expression
1 equsalhwOLD.2 . . . . 5 (x = y → (φψ))
2 sp 1747 . . . . . 6 (xψψ)
3 equsalhwOLD.1 . . . . . 6 (ψxψ)
42, 3impbii 180 . . . . 5 (xψψ)
51, 4syl6bbr 254 . . . 4 (x = y → (φxψ))
65pm5.74i 236 . . 3 ((x = yφ) ↔ (x = yxψ))
76albii 1566 . 2 (x(x = yφ) ↔ x(x = yxψ))
83a1d 22 . . . 4 (ψ → (x = yxψ))
93, 8alrimih 1565 . . 3 (ψx(x = yxψ))
10 ax9v 1655 . . . . 5 ¬ x ¬ x = y
11 con3 126 . . . . . 6 ((x = yxψ) → (¬ xψ → ¬ x = y))
1211al2imi 1561 . . . . 5 (x(x = yxψ) → (x ¬ xψx ¬ x = y))
1310, 12mtoi 169 . . . 4 (x(x = yxψ) → ¬ x ¬ xψ)
14 ax6o 1750 . . . 4 x ¬ xψψ)
1513, 14syl 15 . . 3 (x(x = yxψ) → ψ)
169, 15impbii 180 . 2 (ψx(x = yxψ))
177, 16bitr4i 243 1 (x(x = yφ) ↔ ψ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540 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-11 1746 This theorem depends on definitions:  df-bi 177  df-ex 1542 This theorem is referenced by: (None)
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